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STAT2 
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ON  ELEMENTARY  MATHEMATICS 


IN  THE  SAME  SERIES. 


ON  CONTINUITY  AND  IRRATIONAL  NUMBERS,  and 
ON  THE  NATURE  AND  MEANING  OF  NUMBERS. 
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GEOMETRIC  EXERCISES  IN  PAPER-FOLDING.  By  T. 
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net  (45.  6d.  net). 

LECTURES  ON  ELEMENTARY  MATHEMATICS.  By 
JOSEPH  Louis  LAGRANGE.  From  the  French  by  Thomas  J. 
McCormack.  With  portrait  and  biography.  Pages,  172. 
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ELEMENTARY  ILLUSTRATIONS  OF  THE  DIFFEREN- 
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A  BRIEF  HISTORY  OF  ELEMENTARY  MATHEMATICS. 
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THE  OPEN  COURT  PUBLISHING  COMPANY 

324    DEARBORN    ST.,    CHICAGO. 

LONDON:  Kegan  Paul,  Trench,  Trubner  &  Co. 


LECTURES 


ELEMENTARY  MATHEMATICS 


JOSEPH  LOUIS  LAGRANGE 


FROM  THE  FRENCH  BY 


THOMAS  J.  McCORMACK 

- 


/  5  3  b  0 

SECOND  EDITION 


CHICAGO 

THE  OPEN  COURT  PUBLISHING  COMPANY 

LONDON  AGENTS 

KEGAN  PAUL,  TRENCH,  TRUBNER  &  Co.,  LTD, 
1901 


UN     1907 


TRANSLATION  COPYRIGHTED 
BY 

THE  OPEN  COURT  PUBLISHING  Co. 
1808. 


Engineer  ing  & 
Mathematical 

Sciences 

Library 


^v 


PREFACE. 


THE  present  work,  which  is  a  translation  of  the  Lecons  ele- 
mentaires  sur  les  mathematiques  of  Joseph  Louis  Lagrange, 
the  greatest  of  modern  analysts,  and  which  is  to  be  found  in  Vol- 
ume VII.  of  the  new  edition  of  his  collected  works,  consists  of  a 
series  of  lectures  delivered  in  the  year  1795  at  the  Ecole  ATormale, 
— an  institution  which  was  the  direct  outcome  of  the  French  Re- 
volution and  which  gave  the  first  impulse  to  modern  practical 
ideals  of  education.  With  Lagrange,  at  this  institution,  were  asso- 
ciated, as  professors  of  mathematics.  Monge  and  Laplace,  and  we 
owe  to  the  same  historical  event  the  final  form  of  the  famous  Geo- 
metric descriptive,  as  well  as  a  second  course  of  lectures  on  arith- 
metic and  algebra,  introductory  to  these  of  Lagrange,  by  Laplace. 

With  the  exception  of  a  German  translation  by  Niedermiiller 
(Leipsic,  1880),  the  lectures  of  Lagrange  have  never  been  pub- 
lished in  separate  form  ;  originally  they  appeared  in  a  fragmentary 
shape  in  the  Seances  des  Ecoles  Normales,  as  they  had  been  re- 
ported by  the  stenographers,  and  were  subsequently  reprinted  in 
the  journal  of  the  Polytechnic  School.  From  references  in  them 
to  subjects  afterwards  to  be  treated  it  is  to  be  inferred  that  a  fuller 
development  of  higher  algebra  was  intended, — an  intention  which 
the  brief  existence  of  the  Ecole  Normale  defeated.  With  very  few 
exceptions,  we  have  left  the  expositions  in  their  historical  form, 
having  only  referred  in  an  Appendix  to  a  point  in  the  early  history 
of  algebra. 

The  originality,  elegance,  and  symmetrical  character  of  these 
lectures  have  been  pointed  out  by  DeMorgan,  and  notably  by  Diih- 
ring,  who  places  them  in  the  front  rank  of  elementary  expositions, 
as  an  exemplar  of  their  kind.  Coming,  as  they  do,  from  one  of 
the  greatest  mathematiciar.s  of  modern  times,  and  with  all  the  ex- 
cellencies which  such  a  source  implies,  unique  in  their  character 


VI  PREFACE. 

as  a  reading-book  in  mathematics,  and  interwoven  with  historical 
and  philosophical  remarks  of  great  helpfulness,  they  cannot  fail 
to  have  a  beneficent  and  stimulating  influence, 

The  thanks  of  the  translator  of  the  present  volume  are  due  to 
Professor  Henry  B.  Fine,  of  Princeton,  N.  J.,  for  having  read  the 
proofs. 

THOMAS  J.  McCoRMACK. 
LA  SALLE,  ILLINOIS,  August  i,  1898. 


JOSEPH  LOUIS  LAGRANGE. 

A  , 

«• 

BIOGRAPHICAL   SKETCH. 

/  6  2>  b  O 
\    GREAT  part  of  the  progress  of  formal  thought,  where  it  is 

7  not  hampered  by  outward  causes,  has  been  due  to  the  inven- 
tion of  what  we  may  call  stenophrenic,  or  short-mind,  symbols. 
These,  of  which  all  written  language  and  scientific  notations  are 
examples,  disengage  the  mind  from  the  consideration  of  ponderous 
and  circuitous  mechanical  operations  and  economise  its  energies 
for  the  performance  of  new  and  unaccomplished  tasks  of  thought. 
And  the  advancement  of  those  sciences  has  been  most  notable 
%vhich  have  made  the  most  extensive  use  of  these  short-mind  sym- 
bols. Here  mathematics  and  chemistry  stand  pre-eminent.  The 
ancient  Greeks,  with  all  their  mathematical  endowment  as  a  race, 
and  even  admitting  that  their  powers  were  more  visualistic  than 
analytic,  were  yet  so  impeded  by  their  lack  of  short-mind  symbols 
as  to  have  made  scarcely  any  progress  whatever  in  analysis.  Their 
arithmetic  was  a  species  of  geometry.  They  did  not  possess  the 
sign  for  zero,  and  also  did  not  make  use  of  position  as  an  indicator 
of  value.  Even  later,  when  the  germs  of  the  indeterminate  ana'y- 
sis  were  disseminated  in  Europe  by  Diophantus,  progress  ceased 
here  in  the  science,  doubtless  from  this  very  cause.  The  histori- 
cal calculations  of  Archimedes,  his  approximation  to  the  value  of 
77,  etc  ,  owing  to  this  lack  of  appropriate  arithmetical  and  algebra- 
ical symbols,  entailed  enormous  and  incredible  labors,  which,  if 
they  had  been  avoided,  would,  with  his  genius,  indubitably  have 
led  to  great  discoveries. 


Vlll  BIOGRAPHICAL  SKETCH. 

Subsequently,  at  the  close  of  the  Middle  Ages,  when  the  so- 
called  Arabic  figures  became  established  throughout  Europe  with 
the  symbol  0  and  the  principle  of  local  value,  immediate  progress 
was  made  in  the  art  of  reckoning.  The  problems  which  arose 
gave  rise  to  questions  of  increasing  complexity  and  led  up  to  the 
general  solutions  of  equations  of  the  third  and  fourth  degree  by 
the  Italian  mathematicians  of  the  sixteenth  century.  Yet  even 
these  discoveries  were  made  in  somewhat  the  same  manner  as 
problems  in  mental  arithmetic  are  now  solved  in  common  schools ; 
for  the  present  signs  of  plus,  minus,  and  equality,  the  radical  and 
exponential  signs,  and  especially  the  systematic  use  of  letters  for 
denoting  general  quantities  in  algebra,  had  not  yet  become  univer- 
sal. The  last  step  was  definitively  due  to  the  French  mathema- 
tician Vieta  (1540-1603),  and  the  mighty  advancement  of  analysis 
resulting  therefrom  can  hardly  be  measured  or  imagined.  The 
trammels  were  here  removed  from  algebraic  thought,  and  it  ever 
afterwards  pursued  its  way  unincumbered  in  development  as  if  im- 
pelled by  some  intrinsic  and  irresistible  potency.  Then  followed 
the  introduction  of  exponents  by  Descartes,  the  representation  of 
geometrical  magnitudes  by  algebraical  symbols,  the  extension  of 
the  theory  of  exponents  to  fractional  and  negative  numbers  by 
Wallis  (1616-1703),  and  other  symbolic  artifices,  which  rendered 
the  language  of  analysis  as  economic,  unequivocal,  and  appropriate 
as  the  needs  of  the  science  appeared  to  demand.  In  the  famous 
dispute  regarding  the  invention  of  the  infinitesimal  calculus,  while 
not  denying  and  even  granting  for  the  nonce  the  priority  of  Newton 
in  the  matter,  some  writers  have  gone  so  far  as  to  regard  Leibnitz's 
introduction  of  the  integral  symbol  f  as  alone  a  sufficient  substan- 
tiation of  his  claims  to  originality  and  independence,  so  far  as  the 
power  of  the  new  science  was  concerned. 

For  the  development  of  science  all  such  short-mind  symbols 
are  of  paramount  importance,  and  seem  to  carry  within  themselves 
the  germ  of  a  perpetual  mental  motion  which  needs  no  outward 
power  for  its  unfoldment.  Euler's  well-known  saying  that  his 


BIOGRAPHICAL  SKETCH.  IX 

pencil  seemed  to  surpass  him  in  intelligence  finds  its  explanation 
here,  and  will  be  understood  by  all  who  have  experienced  the  un- 
canny feeling  attending  the  rapid  development  of  algebraical  form- 
ulae, where  the  urned  thought  of  centuries,  so  to  speak,  rolls  from 
one's  finger's  ends. 

But  it  should  never  be  forgotten  that  the  mighty  stenophrenic 
engine  of  which  we  here  speak,  like  all  machinery,  affords  us  rather 
a  mastery  over  nature  than  an  insight  into  it ;  and  for  some,  un- 
fortunately, the  higher  symbols  of  mathematics  are  merely  bram- 
bles that  hide  the  living  springs  of  reality.  Many  of  the  greatest 
discoveries  of  science, — for  example,  those  of  Galileo,  Huygens, 
and  Newton, — were  made  without  the  mechanism  which  afterwards 
becomes  so  indispensable  for  their  development  and  application. 
Galileo's  reasoning  anent  the  summation  of  the  impulses  imparted 
to  a  falling  stone  is  virtual  integration  ;  and  Newton's  mechanical 
discoveries  were  made  by  the  man  who  invented,  but  evidently  did 
not  use  to  that  end,  the  doctrine  of  fluxions. 

* 

We  have  been  following  here,  briefly  and  roughly,  a  line  of 
progressive  abstraction  and  generalisation  which  even  in  its  begin- 
ning was,  psychologically  speaking,  at  an  exalted  height,  but  in  the 
course  of  centuries  had  been  carried  to  points  of  literally  ethereal 
refinement  and  altitude.  In  that  long  succession  of  inquirers  by 
whom  this  result  was  effected,  the  process  reached,  we  may  say, 
its  culmination  and  purest  expression  in  Joseph  Louis  Lagrange, 
born  in  Turin,  Italy,  the  soth  of  January,  1736,  died  in  Paris,  April 
10,  1813.  Lagrange's  power  over  symbols  has,  perhaps,  never  been 
paralleled  either  before  his  day  or  since.  It  is  amusing  to  hear  his 
biographers  relate  that  in  early  life  he  evinced  no  aptitude  for 
mathematics,  but  seemed  to  have  been  given  over  entirely  to  the 
pursuits  of  pure  literature  ;  for  at  fifteen  we  find  him  teaching 
mathematics  in  an  artillery  school  in  Turin,  and  at  nineteen  he 
had  made  the  greatest  discovery  in  mathematical  science  since  that 
of  the  infinitesimal  calculus,  namely,  the  creation  of  the  algorism 


X  BIOGRAPHICAL  SKETCH. 

and  method  of  the  Calculus  of  Variations.  ' '  Your  analytical  so- 
lution of  the  isoperimetrical  problem,"  writes  Euler,  then  the  prince 
of  European  mathematicians,  to  him,  "  leaves  nothing  to  be  desired 
in  this  department  of  inquiry,  and  I  am  delighted  beyond  measure 
that  it  has  been  your  lot  to  carry  to  the  highest  pitch  of  perfection, 
a  theory,  which  since  its  inception  I  have  been  almost  the  only  one 
to  cultivate."  But  the  exact  nature  of  a  "variation"  even  Euler 
did  not  grasp,  and  even  as  late  as  1810  in  the  English  treatise  of 
Woodhouse  on  this  subject  we  read  regarding  a  certain  new  sign 
introduced,  that  M.  Lagrange's  "power  over  symbols  is  so  un- 
bounded that  the  possession  of  it  seems  to  have  made  him  capri- 
cious." 

Lagrange  himself  was  conscious  of  his  wonderful  capacities  in 
this  direction.  His  was  a  time  when  geometry,  as  he  himself 
phrased  it,  had  become  a  dead  language,  the  abstractions  of  analy- 
sis were  being  pushed  to  their  highest  pitch,  and  he  felt  that  with 
his  achievements  its  possibilities  within  certain  limits  were  being 
rapidly  exhausted.  The  saying  is  attributed  to  him  that  chairs  of 
mathematics,  so  far  as  creation  was  concerned,  and  unless  new 
fields  were  opened  up,  would  soon  be  as  rare  at  universities  as 
chairs  of  Arabic.  In  both  research  and  exposition,  he  totally  re- 
versed the  methods  of  his  predecessors.  They  had  proceeded  in 
their  exposition  from  special  cases  by  a  species  of  induction  ;  his 
eye  was  always  directed  to  the  highest  and  most  general  points  of 
view  ;  and  it  was  by  his  suppression  of  details  and  neglect  of  minor, 
unimportant  considerations  that  he  swept  the  whole  field  of  anal- 
ysis with  a  generality  of  insight  and  power  never  excelled,  adding 
to  his  originality  and  profundity  a  conciseness,  elegance,  and  lu- 
cidity which  have  made  him  the  model  of  mathematical  writers. 

* 

Lagrange  came  of  an  old  French  family  of  Touraine,  France, 
said  to  have  been  allied  to  that  of  Descartes.  At  the  age  of  twenty- 
six  he  found  himself  at  the  zenith  of  European  fame.  But  his 
reputation  had  been  purchased  at  a  great  cost.  Although  of  ordi- 


BIOGRAPHICAL  SKETCH.  XI 

nary  height  and  well  proportioned,  he  had  by  his  ecstatic  devotion 
to  study, — periods  always  accompanied  by  an  irregular  pulse  and 
high  febrile  excitatian, — almost  ruined  his  health.  At  this  age, 
accordingly,  he  was  seized  with  a  hypochondriacal  affection  and 
with  bilious  disorders,  which  accompanied  him  thronghout  his  life, 
and  which  were  only  allayed  by  his  great  abstemiousness  and  care- 
ful regimen.  He  was  bled  twenty-nine  times,  an  infliction  which 
alone  would  have  affected  the  most  robust  constitution.  Through 
his  great  care  for  his  health  he  gave  much  attention  to  medicine. 
He  was,  in  fact,  conversant  with  all  the  sciences,  although  know- 
ing his  forte  he  rarely  expressed  an  opinion  on  anything  uncon- 
nected with  mathematics. 

When  Euler  left  Berlin  for  St.  Petersburg  in  1766  he  and 
D'Alembert  induced  Frederick  the  Great  to  make  Lagrange  presi- 
dent of  the  Academy  of  Sciences  at  Berlin.  Lagrange  accepted 
the  position  and  lived  in  Berlin  twenty  years,  where  he  wrote  some 
of  his  greatest  works.  He  was  a  great  favorite  of  the  Berlin  peo- 
ple, and  enjoyed  the  profoundest  respect  of  Frederick  the  Great, 
although  the  latter  seems  to  have  preferred  the  noisy  reputation  of 
Maupertuis,  Lamettrie,  and  Voltaire  to  the  unobtrusive  fame  and 
personality  of  the  man  whose  achievements  were  destined  to  shed 
more  lasting  light  on  his  reign  than  those  of  any  of  his  more  strident 
literary  predecessors  :  Lagrange  was,  as  he  himself  said,  fhilosophe 
sans  crier. 

The  climate  of  Prussia  agreed  with  the  mathematician.  He 
refused  the  most  seductive  offers  of  foreign  courts  and  princes,  and 
it  was  not  until  the  death  of  Frederick  and  the  intellectual  reaction 
of  the  Prussian  court  that  he  returned  to  Paris,  where  his  career 
broke  forth  in  renewed  splendor.  He  published  in  1788  his  great 
Mecanique  analytique,  that  "scientific  poem"  of  Sir  William 
Rowan  Hamilton,  which  gave  the  quietus  to  mechanics  as  then 
formulated,  and  having  been  made  during  the  Revolution  Profes- 
sor of  Mathematics  at  the  new  Ecole  Normale  and  the  Ecole  Poly- 
technique,  he  entered  with  Laplace  and  Monge  upon  the  activity 


Xll  BIOGRAPHICAL  SKETCH. 

which  made  these  schools  for  generations  to  come  exemplars  of 
practical  scientific  education,  systematising  by  his  lectures  there, 
and  putting  into  definitive  form,  the  science  of  mathematical  anal- 
ysis of  which  he  had  developed  the  extremest  capacities.  La- 
grange's  activity  at  Paris  was  interrupted  only  once  by  a  brief  pe- 
riod of  melancholy  aversion  for  mathematics,  a  lull  which  he 
devoted  to  the  adolescent  science  of  chemistry  and  to  philosophical 
studies  ;  but  he  afterwards  resumed  his  old  love  with  increased  ar- 
dor and  assiduity.  His  significance  for  thought  generally  is  far 
beyond  what  we  have  space  to  insist  upon.  Not  least  of  all,  theol- 
ogy, which  had  invariably  mingled  itself  with  the  researches  of  his 
predecessors,  was  with  him  forever  divorced  from  a  legitimate  in- 
fluence of  science. 

The  honors  of  the  world  sat  ill  upon  Lagrange  :  la  magnifi- 
cence le  genait,  he  said ;  but  he  lived  at  a  time  when  proffered 
things  were  usually  accepted,  not  refused.  He  was  loaded  with 
personal  favors  and  official  distinctions  by  Napoleon  who  called 
him  la  haute  pyramide  des  sciences  mathematiques ,  was  made  a 
Senator,  a  Count  of  the  Empire,  a  Grand  Officer  of  the  Legion  of 
Honor,  and,  just  before  his  death,  received  the  grand  cross  of  the 
Order  of  Reunion.  He  never  feared  death,  which  he  termed  une 
dernier e  function,  ni  pem'ble  ni  dtsagreable,  much  less  the  dis- 
approval of  the  great.  He  remained  in  Paris  during  the  Revolu- 
tion when  savants  were  decidedly  in  disfavor,  but  was  suspected 
of  aspiring  to  no  throne  but  that  of  mathematics.  When  Lavoisier 
was  executed  he  said :  "It  took  them  but  a  moment  to  lay  low  that 
head  ;  yet  a  hundred  years  will  not  suffice  perhaps  to  produce  its 
like  again." 

Lagrange  would  never  allow  his  portrait  to  be  painted,  main- 
taining that  a  man's  works  and  not  his  personality  deserved  pre- 
servation. The  frontispiece  to  the  present  work  is  from  a  steel 
engraving  based  on  a  sketch  obtained  by  stealth  at  a  meeting  of 
the  Institute.  His  genius  was  excelled  only  by  the  purity  and 
nobleness  of  his  character,  in  which  the  world  never  even  sought 


BIOGRAPHICAL  SKETCH.  Xlll 

to  find  a  blot,  and  by  the  exalted  Pythagorean  simplicity  of  his 
life.  He  was  twice  married,  and  by  his  wonderful  care  of  his  per- 
son lived  to  the  advanced  age  of  seventy-seven  years,  not  one  of 
which  had  been  misspent.  His  life  was  the  veriest  incarnation  of 
the  scientific  spirit ;  he  lived  for  nothing  else.  He  left  his  weak 
body,  which  retained  its  intellectual  powers  to  the  very  last,  "as  an 
offering  upon  the  altar  of  science,  happily  made  when  his  work 
had  been  done  ;  but  to  the  world  he  bequeathed  his  "  ever-living" 
thoughts  now  recently  resurgent  in  a  new  and  monumental  edition 
of  his  works  (published  by  Gauthier-Villars,  Paris).  Ma  vie  est 
la!  he  said,  pointing  to  his  brain  the  day  before  his  death. 

THOMAS  J.  McCoRMACK. 


CONTENTS. 


PREFACE    

BIOGRAPHICAL  SKETCH  OF  JOSEPH  Louis  LAGRANGE. 

LECTURE  I.  ON  ARITHMETIC,  AND  IN  PARTICULAR  FRAC- 
TIONS AND  LOGARITHMS 1-23 

Systems  of  Numeration. — Fractions. — Greatest  Com- 
mon Divisor. —  Continued  Fractions. — Theory  of 
Powers,  Proportions,  and  Progressions. — Involution 
and  Evolution. — -Rule  of  Three. — Interest. — Annui- 
ties. — Logarithms. 

LECTURE  II.  ON  THE  OPERATIONS  OF  ARITHMETIC  .  .  .  24-53 
Arithmetic  and  Geometry. — New  Method  of  Sub- 
traction.— Abridged  and  Approximate  Multiplica- 
tion.—  Decimals. — Property  of  the  Number  9. — 
Tests  of  Divisibility. — Theory  of  Remainders. — 
Checks  on  Multiplication  and  Division. — Evolution. 
— Rule  of  Three. — Theory  and  Practice. — Probabil- 
ity of  Life. — Alligation  or  the  Rule  of  Mixtures. 

LECTURE  III.  ON  ALGEBRA,  PARTICULARLY  THE  RESOLU- 
TION OF  EQUATIONS  OF  THE  THIRD  AND  FOURTH  DE- 
GREE    54-95 

Origin  of  Greek  Algebra. — Diophantus. — Indetermi- 
nate Analysis. — Equations  of  the  Second  Degree. — 
Translations  of  Diophantus. — Algebra  Among  the 
Arabs. — History  of  Algebra  in  Italy,  France,  and 
Germany. — History  of  Equations  of  the  Third  and 
Fourth  Degree  and  of  the  Irreducible  Case. —The- 
ory of  Equations- — Discussion  of  Cubic  Equations. 
— Discussion  of  the  Irreducible  Case — The  Theory 


XVI  CONTENTS. 

PAGES 

of  Roots.— Extraction  of  the  Square  and  Cube  Roots 
of  Two  Imaginary  Binomials. — Theory  of  Imagin- 
ary Expressions. — Trisection  of  an  Angle. — Method 
of  Indeterminates. — Discussion  of  Biquadratic  Equa- 
tions. 

LECTURE  IV.     ON  THE  RESOLUTION  OF  NUMERICAL  EQUA- 
TIONS    96-126 

Algebraical  Resolution  of  Equations.— Numerical 
Resolution  of  Equations. — Position  of  the  Roots. — 
Representation  of  Equations  by  Curves. — Graphic 
Resolution  of  Equations. — Character  of  the  Roots  of 
Equations. — Limits  of  the  Roots  of  Numerical  Equa- 
tions.— Separation  of  the  Roots. — Method  of  Substi- 
tutions.— The  Equation  of  Differences. — Method  of 
Elimination. — Constructions  and  Instruments  for 
Solving  Equations. 

LECTURE  V.     ON  THE  EMPLOYMENT  OF  CURVES  IN  THE  SO- 
LUTION OF  PROBLEMS 127-149 

Application  of  Geometry  to  Algebra. — Resolution  of 
Problems  by  Curves. — The  Problem  of  Two  Lights. 
— Variable  Quantities — Minimal  Values. — Analysis 
of  Biquadratic  Equations  Conformably  to  the  Prob- 
lem of  the  Two  Lights. — Advantages  of  the  Method 
of  Curves — The  Curve  of  Errors. — Regulafalsi. — 
Solution  of  Problems  by  the  Curve  of  Errors. — 
Problem  of  the  Circle  and  Inscribed  Polygon. — 
Problem  of  the  Observer  and  Three  Objects. — Par- 
abolic Curves. — Newton's  Problem. — Interpolation 
of  Intermediate  Terms  in  Series  of  Observations, 
Experiments,  etc. 

APPENDIX 151 

Note  on  the  Origin  of  Algebra. 


LECTURE  I. 

ON    ARITHMETIC,    AND   IN    PARTICULAR    FRACTIONS 
AND  LOGARITHMS, 

A  RITHMETIC  is  divided  into  two  parts.    The  first 
•**•     is  based  on  the  decimal  system  of  notation  and  systems  of 

,  r  •  11  numeration 

on  the  manner  of  arranging  numeral  characters  to  ex- 
press numbers.  This  first  part  comprises  the  four 
common  operations  of  addition,  subtraction,  multi- 
plication, and  division, — operations  which,  as  you 
know,  would  be  different  if  a  different  system  were 
adopted,  but,  which  it  would  not  be  difficult  to  trans- 
form from  one  system  to  another,  if  a  change  of  sys- 
tems were  desirable. 

The  second  part  is  independent  of  the  system  of 
numeration.  It  is  based  on  the  consideration  of  quan- 
tities and  on  the  general  properties  of  numbers.  The 
theory  of  fractions,  the  theory  of  powers  and  of  roots, 
the  theory  of  arithmetical  and  geometrical  progres- 
sions, and,  lastly,  the  theory  of  logarithms,  fall  under 
this  head.  I  purpose  to  advance,  here,  some  remarks 
on  the  different  branches  of  this  part  of  arithmetic. 


2  ON  ARITHMETIC. 

It  may  be  regarded  as  universal  arithmetic,  having 
an  intimate  affinity  to  algebra.  For,  if  instead  of 
particularising  the  quantities  considered,  if  instead  of 
assigning  them  numerically,  we  treat  them  in  quite  a 
general  way,  designating  them  by  letters,  we  have 
algebra. 

You  know  what  a  fraction  is.  The  notion  of  a 
Fractions,  fraction  is  slightly  more  composite  than  that  of  whole 
numbers.  In  whole  numbers  we  consider  simply  a 
quantity  repeated.  To  reach  the  notion  of  a  fraction 
it  is  necessary  to  consider  the  quantity  divided  into  a 
certain  number  of  parts.  Fractions  represent  in  gen- 
eral ratios,  and  serve  to  express  one  quantity  by  means 
of  another.  In  general,  nothing  measurable  can  be 
measured  except  by  fractions  expressing  the  result  of 
the  measurement,  unless  the  measure  be  contained  an 
exact  number  of  times  in  the  thing  to  be  measured. 

You  also  know  how  a  fraction  can  be  reduced  to 
its  lowest  terms.  When  the  numerator  and  the  de- 
nominator are  both  divisible  by  the  same  number, 
their  greatest  common  divisor  can  be  found  by  a  very 
ingenious  method  which  we  owe  to  Euclid.  This 
method  is  exceedingly  simple  and  lucid,  but  it  may 
be  rendered  even  more  palpable  to  the  eye  by  the  fol- 
lowing consideration.  Suppose,  for  example,  that  you 
have  a  given  length,  and  that  you  wish  to  measure  it. 
The  unit  of  measure  is  given,  and  you  wish  to  know 
how  many  times  it  is  contained  in  the  length.  You 
first  lay  off  your  measure  as  many  times  as  you  can  on. 


ON  ARITHMETIC.  3 

the  given  length,  and  that  gives  you  a  certain  whole 
number  of  measures.  If  there  is  no  remainder  your 
operation  is  finished.  But  if  there  be  a  remainder,  Greatest 
that  remainder  is  still  to  be  evaluated.  If  the  meas-  dh^on" 
ure  is  divided  into  equal  parts,  for  example,  into  ten, 
twelve,  or  more  equal  parts,  the  natural  procedure  is 
to  use  one  of  these  parts  as  a  new  measure  and  to  see 
how  many  times  it  is  contained  in  the  remainder. 
You  will  then  have  for  the  value  of  your  remainder, 
a  fraction  of  which  the  numerator  is  the  number  of 
parts  contained  in  the  remainder  and  the  denominator 
the  total  number  of  parts  into  which  the  given  meas- 
ure is  divided. 

I  will  suppose,  now,  that  your  measure  is  not  so 
divided  but  that  you  still  wish  to  determine  the  ratio 
of  the  proposed  length  to  the  length  which  you  have 
adopted  as  your  measure.  The  following  is  the  pro- 
cedure which  most  naturally  suggests  itself. 

If  you  have  a  remainder,  since  that  is  less  than  the 
measure,  naturally  you  will  seek  to  find  how  many 
•times  your  remainder  is  contained  in  this  measure. 
Let  us  say  two  times,  and  that  a  remainder  is  still 
left.  Lay  this  remainder  on  the  preceding  remainder. 
Since  it  is  necessarily  smaller,  it  will  still  be  contained 
a  certain  number  of  times  in  the  preceding  remainder, 
say  three  times,  and  there  will  be  another  remainder 
or  there  will  not  ;  and  so  on.  In  these  different  re- 
mainders you  will  have  what  is  called  a  continued  frac- 
tion. For  example,  you  have  found  that  the  measure 


£  ON  ARITHMETIC. 

is  contained  three  times  in  the  proposed  length.  You 
have,  to  start  with,  the  number  three.  Then  you  have 
continued  found  that  your  first  remainder  is  contained  twice  in 
your  measure.  You  will  have  the  fraction  one  divided 
by  two.  But  this  last  denominator  is  not  .complete. 
for  it  was  supposed  there  was  still  a  remainder.  That 
remainder  will  give  another  and  similar  fraction,  which 
is  to  be  added  to  the  last  denominator,  and  which  by 
our  supposition  is  one  divided  by  three.  And  so  with 
the  rest.  You  will  then  have  the  fraction 


2  +  1 


as  the  expression  of  your  ratio  between  the  proposed 
length  and  the  adopted  measure. 

Fractions  of  this  form  are  called  continued  fractions, 
and  can  be  reduced  to  ordinary  fractions  by  the  com- 
mon rules.  Thus,  if  we  stop  at  the  first  fraction,  i.  e. , 
if  we  consider  only  the  first  remainder  and  neglect  the 
second,  we  shall  have  3-|-  ^,  which  is  equal  to  \.  Con- 
sidering only  the  first  and  the  second  remainders,  we 

stop  at  the  second  fraction,  and  shall  have  3-)-— 

Now  2  -f  i  =  |.  We  shall  have  therefore  3  -f-  ^,  which 
is  equal  to  -2T4-.  And  so  on  with  the  rest.  If  we  arrive 
in  the  course  of  the  operation  at  a  remainder  which  is 
contained  exactly  in  the  preceding  remainder,  the 
operation  is  terminated,  and  we  shall  have  in  the  con- 


•ac- 
tions. 


ON  ARITHMETIC.  5 

tinued  fraction  a  common  fraction  that  is  the  exact 
value  of  the  length  to  be  measured,  in  terms  of  the 
length  which  served  as  our  measure.    If  the  operation  Terminat- 
is  not  thus  terminated,  it  can  be  continued  to  infinity,  ued  fra 
and  we  shall  have  only  fractions  which  approach  more 
and  more  nearly  to  the  true  value. 

If  we  now  compare  this  procedure  with  that  em- 
ployed for  finding  the  greatest  common  divisor  of  two 
numbers,  we  shall  see  that  it  is  virtually  the  same 
thing ;  the  difference  being  that  in  finding  the  great- 
est common  divisor  we  devote  our  attention  solely  to 
the  different  remainders,  of  which  the  last  is  the  di- 
visor sought,  whereas  by  employing  the  successive 
quotients,  as  we  have  done  above,  we  obtain  fractions 
which  constantly  approach  nearer  and  nearer  to  the 
fraction  formed  by  the  two  numbers  given,  and  of 
which  the  last  is  that  fraction  itself  reduced  to  its 
lowest  terms. 

As  the  theory  of  continued  fractions  is  little  known, 
but  is  yet  of  great  utility  in  the  solution  of  impor- 
tant numerical  questions,  I  shall  enter  here  somewhat 
more  fully  into  the  formation  and  properties  of  these 
fractions.  And,  first,  let  us  suppose  that  the  quotients 
found,  whether  by  the  mechanical  operation,  or  by 
the  method  for  finding  the  greatest  common  divisor, 
are,  as  above,  3,  2,  3,  5,  7,  3.  The  following  is  a  rule 
by  which  we  can  write  down  at  once  the  convergent 
fractions  which  result  from  these  quotients,  without 
developing  the  continued  fraction. 


O  ON  ARITHMETIC. 

'The   first   quotient,    supposed    divided    by   unity. 

will  give  the  first  fraction,  which  will  be  too  small, 

Converging  namely,  £.      Then,  multiplying  the  numerator  and  de- 

Iractions.  . 

nominator  of  this  fraction  by  the  second  quotient  and 
adding  unity  to  the  numerator,  we  shall  have  the  sec- 
ond fraction,  •£,  which  will  be  too  large.  Multiplying 
in  like  manner  the  numerator  and  denominator  of  this 
fraction  by  the  third  quotient,  and  adding  to  the  nu- 
merator the  numerator  of  the  preceding  fraction,  and 
to  the  denominator  the  denominator  of  the  preceding 
fraction,  we  shall  have  the  third  fraction,  which  will 
be  too  small.  Thus,  the  third  quotient  being  3,  we 
have  for  our  numerator  (7  X  3  =  21)-}- 3  =  24,  and  for 
our  denominator  (2  X  3  =  6)-|- 1  =  7.  The  third  con- 
vergent, therefore,  is  %f-.  We  proceed  in  the  same 
manner  for  the  fourth  convergent.  The  fourth  quo- 
tient being  5,  we  say  24  times  5  is  120,  and  this  plus 
7,  the  numerator  of  the  fraction  preceding,  is  127  ; 
similarly,  7  times  5  is  35,  and  this  plus  2  is  37.  The 
new  fraction,  therefore,  is  Jy77-.  And  so  with  the  rest. 
In  this  manner,  by  employing  the  six  quotients  3, 
2,  3,  5,  7,  3  we  obtain  the  six  fractions 

3      7      24      127       913      2866 
T'     2"'    T'    "37"'    "266'    ~835~' 

of  which  the  last,  supposing  the  operation  to  be  com- 
pleted at  the  sixth  quotient  3,  will  be  the  required 
value  of  the  length  measured,  or  the  fraction  itself 
reduced  to  its  lowest  terms. 

The  fractions  which  precede  the  last  are  alternately 


ON  ARITHMETIC.  7 

smaller  and  larger  than  the  last,  and  have  the  advan- 
tage of  approaching  more  and  more  nearly  to  its  value 
in  such  wise  that  no  other  fraction  can  approach  it  Conver- 

fients. 

more  nearly  except  its  denominator  be  larger  than  the 
product  of  the  denominator  of  the  fraction  in  question 
and  the  denominator  of  the  fraction  following.  For 
example,  the  fraction  -274-  is  less  than  the  true  value 
which  is  that  of  the  fraction  ?^//-,  but  it  approaches 
to  it  more  nearly  than  any  other  fraction  does  whose 
denominator  is  not  greater  than  the  product  of  7  by 
37,  that  is,  259.  Thus,  any  fraction  expressed  in  large 
numbers  may  be  reduced  to  a  series  of  fractions  ex- 
pressed in  smaller  numbers  and  which  approach  as 
near  to  it  as  possible  in  value. 

The  demonstration  of  the  foregoing  properties  is 
deduced  from  the  nature  of  continued  fractions,  and 
from  the  fact  that  if  we  seek  the  difference  between 
one  of  the  convergent  fractions  and  that  next  adjacent 
to  it  we  shall  obtain  a  fraction  of  which  the  numerator 
is  always  unity  and  the  denominator  the  product  of 
the  two  denominators;  a  consequence  which  follows 
a  priori  from  the  very  law  of  formation  of  these  frac- 
tions. Thus  the  difference  between  £  and  \  is  ^,  in 
excess  ;  between  -2T4-  and  %,  -^,  in  defect ;  between  Jj277- 
and  £=*-,  ^i^,  in  excess ;  and  so  on.  The  result  being, 
that  by  employing  this  series  of  differences  we  can 
express  in  another  and  very  simple  manner  the  frac- 
tions with  which  we  are  here  concerned,  by  means  of 
a  second  series  of  fractions  of  which  the  numerators 


8  ON  ARITHMETIC. 

are  all  unity  and  the  denominators  successively  the 

products  of   every  two   adjacent  denominators.      In- 

A  second      stead  of  the  fractions  written  above,  we  have  thus  the 

method  of 
expression,    series  '. 

311  1  1  1 

I I I 

i      r  i  vx  o       c>  ^  ,  r?    i      n  \/  'JIT         07  ^s  O£i:    i    • 


1    '    1  X  -      2  x  <        7  X  37       37  X  266    '   260  X 

The  first  term,  as  we  see,  is  the  first  fraction,  the 
first  and  second  together  give  the  second  fraction  |, 
the  first,  the  second,  and  the  third  give  the  third  frac- 
tion -2TS  and  so  on  with  the  rest ;  the  result  being  that 
the  series  entire  is  equivalent  to  the  last  fraction. 

There  is  still  another  way,  less  known  but  in  some 
respects  more  simple,  of  treating  the  same  question — 
which  leads  directly  to  a  series  similar  to  the  preced- 
ing. Reverting  to  the  previous  example,  after  having 
found  that  the  measure  goes  three  times  into  the  length 
to  be  measured  and  that  after  the  first  remainder  has 
been  applied  to  the  measure  there  is  left  a  new  re- 
mainder, instead  of  comparing  this  second  remainder 
with  the  preceding,  as  we  did  above,  we  may  compare 
it  with  the  measure  itself.  Thus,  supposing  it  goes 
into  the  latter  seven  times  with  a  remainder,  we  again 
compare  this  last  remainder  with  the  measure,  and  so 
on,  until  we  arrive,  if  possible,  at  a  remainder  which 
is  an  aliquot  part  of  the  measure, — which  will  term- 
inate the  operation.  In  the  contrary  event,  if  the 
measure  and  the  length  to  be  measured  are  incom- 
mensurable, the  process  may  be  continued  to  infinity. 


ON  ARITHMETIC.  9 

We  shall  have  then,  as  the  expression  of  the  length 
measured,  the  series 

J  j  A  third 

3  -(-  - |- .   .    .  method  of 

•^  X    '  expression. 

It  is  clear  that  this  method  is  also  applicable  to 
ordinary  fractions.  We  constantly  retain  the  denomi- 
nator of  the  fraction  as  the  dividend,  and  take  the  dif- 
ferent remainders  successively  as  divisors.  Thus,  the 
fraction  2/gg.  gives  the  quotients  3,  2,  7,  18,  19,  46, 
119,  417  835  ;  from  which  we  obtain  the  series 

31-1          l       I  1  1 

r2       2x7^2x7x18      2x7x18X19 

and  as  these  partial  fractions  rapidly  diminish,  we 
shall  have,  by  combining  them  successively,  the  sim- 
ple fractions, 

7        48  865 

T    2~xT     2x7x18' 

which  will  constantly  approach  nearer  and  nearer  to 
the  true  value  sought,  and  the  error  will  be  less  than 
the  first  of  the  partial  fractions  neglected. 

Our  remarks  on  the  foregoing  methods  of  evaluat- 
ing fractions  should  not  be  construed  as  signifying 
that  the  employment  of  decimal  fractions  is  not  nearly 
always  preferable  for  expressing  the  values  of  fractions 
to  whatever  degree  of  exactness  we  wish.  But  cases 
occur  where  it  is  necessary  that  these  values  should 
be  expressed  by  as  few  figures  as  possible.  For  ex- 
ample, if  it  were  required  to  construct  a  planetarium, 


IO  ON  ARITHMETIC. 

since  the  ratios  of  the  revolutions  of  the  planets  to  one 

another  are  expressed  by  very  large  numbers,  it  would 

origin  of      be   necessary,   in   order   not  to   multiply  unduly   the 

continued 

fractions,  number  of  the  teeth  on  the  wheels,  to  avail  ourselves 
of  smaller  numbers,  but  at  the  same  time  so  to  select 
them  that  their  ratios  should  approach  as  nearly  as 
possible  to  the  actual  ratios.  It  was,  in  fact,  this  very 
question  that  prompted  Huygens,  in  his  search  for  its 
solution,  to  resort  to  continued  fractions  and  that  so 
gave  birth  to  the  theory  of  these  fractions.  After- 
wards, in  the  elaboration  of  this  theory,  it  was  found 
adapted  to  the  solution  of  other  important  questions, 
and  this  is  the  reason,  since  it  is  not  found  in  elemen- 
tary works,  that  I  have  deemed  it  necessary  to  go 
somewhat  into  detail  in  expounding  its  principles. 

We  will  now  pass  to  the  theory  of  powers,  propor- 
tions, and  progressions. 

As  you  already  know,  a  number  multiplied  by  it- 
self gives  its  square,  and  multiplied  again  by  itself 
gives  its  cube,  and  so  on.  In  geometry  we  do  not  go 
beyond  the  cube,  because  no  body  can  have  more  than 
three  dimensions.  But  in  algebra  and  arithmetic  we 
may  go  as  far  as  we  please.  And  here  the  theory  of 
the  extraction  of  roots  takes  its  origin.  For,  although 
every  number  can  be  raised  to  its  square  and  to  its 
cube  and  so  forth,  it  is  not  true  reciprocally  that  every 
number  is  an  exact  square  or  an  exact  cube.  The 
number  2,  for  example,  is  not  a  square  ;  for  the  square 
of  1  is  1,  and  the  square  of  2  is  four  ;  and  there  being 


ON  ARITHMETIC.  II 

no  other  whole  numbers  between  these  two,  it  is  im- 
possible to  find  a  whole  number  which  multiplied  by 
itself  will  give  2.  It  cannot  be  found  in  fractions,  for  involution 

.  and  evolu- 

if  you  take  a  fraction  reduced  to  its  lowest  terms,  the  tion. 
square  of  that  fraction  will  again  be  a  fraction  reduced 
to  its  lowest  terms,  and  consequently  cannot  be  equal 
to  the  whole  number  2.  But  though  we  cannot  obtain 
the  square  root  of  2  exactly,  we  can  yet  approach  to  it 
as  nearly  as  we  please,  particularly  by  decimal  frac- 
tions. By  following  the  common  rules  for  the  extrac- 
tion of  square  roots,  cube  roots,  and  so  forth,  the  pro- 
cess may  be  extended  to  infinity,  and  the  true  values 
of  the  roots  may  be  approximated  to  any  degree  of 
exactitude  we  wish. 

But  I  shall  not  enter  into  details  here.  The  theory 
of  powers  has  given  rise  to  that  of  progressions,  be- 
fore entering  on  which  a  word  is  necessary  on  propor- 
tions. 

Every  fraction  expresses  a  ratio.  Having  two  equal 
fractions,  therefore,  we  have  two  equal  ratios  ;  and 
the  numbers  constituting  the  fractions  or  the  ratios 
form  what  is  called  a  proportion.  Thus  the  equality 
of  the  ratios  2  to  4  and  3  to  6  gives  the  proportion 
2  :  4  :  :  3  :  6,  because  4  is  the  double  of  2  as  G  is  the 
double  of  3.  Many  of  the  rules  of  arithmetic  depend 
on  the  theory  of  proportions.  First,  it  is  the  founda- 
tion of  the  famous  rule  of  three,  which  is  so  extensively 
used.  You  know  that  when  the  first  three  terms  of  a 
proportion  are  given,  to  obtain  the  fourth  you  have 


12  ON  ARITHMETIC. 

only  to  multiply  the  last  two  together  and  divide  the 
product  by  the  first.  Various  special  rules  have  also 
Proportions  been  conceived  and  have  found  a  place  in  the  books 
on  arithmetic  ;  but  they  are  all  reducible  to  the  rule 
of  three  and  may  be  neglected  if  we  once  thoroughly 
grasp  the  conditions  of  the  problem.  There  are  direct, 
inverse,  simple,  and  compound  rules  of  three,  rules  of 
partnership/  of  mixtures,  and  so  forth.  In  all  cases 
it  is  only  necessary  to  consider  carefully  the  condi- 
tions of  the  problem  and  to  arrange  the  terms  of  the 
proportion  correspondingly. 

I  shall  not  enter  into  further  details  here.  There 
is,  however,  another  theory  which  is  useful  on  numer- 
ous occasions, — namely,  the  theory  of  progressions. 
When  you  have  several  numbers  that  bear  the  same 
proportion  to  one  another,  and  which  follow  one  an- 
other in  such  a  manner  that  the  second  is  to  the  first 
as  the  third  is  to  the  second,  as  the  fourth  is  to  the 
third,  and  so  forth,  these  numbers  form  a  progression. 
I  shall  begin  with  an  observation. 

The  books  of  arithmetic  and  algebra  ordinarily  dis- 
tinguish between  two  kinds  of  progression,  arithmet- 
ical and  geometrical,  corresponding  to  the  proportions 
called  arithmetical  and  geometrical.  But  the  appel- 
lation proportion  appears  to  me  extremely  inappro- 
priate as  applied  to  arithmetical  proportion.  And  as  it 
is  one  of  the  objects  of  the  £cole  Normale  to  rectify 
the  language  of  science,  the  present  slight  digression 
will  not  be  considered  irrelevant. 


ON  ARITHMETIC.  13 

I  take  it,  then,  that  the  idea  of  proportion  is  already 
well  established  by  usage  and  that  it  corresponds  solely 
to  what  is  called  geometrical  proportion.  When  we  Arithmet- 

,  .  ical  and 

speak  01  the  proportion  of  the  parts  of  a  man  s  body,  geometri- 
of  the  proportion  of  the  parts  of  an  edifice,  etc. ;  when  ^J 
.we  say  that  a  plan  should  be  reduced  proportionately 
in  size,  etc. ;  in  fact,  when  we  say  generally  that  one 
thing  is  proportional  to  another,  we  understand  by 
proportion  equality  of  ratios  only,  as  in  geometrical 
proportion,  and  never  equality  of  differences  as  in 
arithmetical  proportion.  Therefore,  instead  of  saying 
that  the  numbers,  3,  5,  7,  9,  are  in  arithmetical  pro- 
portion, because  the  difference  between  5  and  3  is  the 
same  as  that  between  9  and  7,  I  deem  it  desirable  that 
some  other  term  should  be  employed,  so  as  to  avoid 
all  ambiguity.  We  might,  for  instance,  call  such  num- 
bers equi  different,  reserving  the  name  of  proportionals 
for  numbers  that  are  in  geometrical  proportion,  as  2, 
4,  6,  8,  etc. 

As  for  the  rest,  I  cannot  see  why  the  proportion 
called  arithmetical  is  any  more  arithmetical  than  that 
which  is  called  geometrical,  nor  why  the  latter  is  more 
geometrical  than  the  former.  On  the  contrary,  the 
primitive  idea  of  geometrical  proportion  is  based  on 
arithmetic,  for  the  notion  of  ratios  springs  essentially 
from  the  consideration  of  numbers. 

Still,  in  waiting  for  these  inappropriate  designa- 
tions to  be  changed,  I  shall  continue  to  make  use  of 
them,  as  a  matter  of  simplicity  and  convenience. 


sions. 


14  ON  ARITHMETIC. 

The  theory  of  arithmetical  progressions  presents 
few  difficulties.  Arithmetical  progressions  consist  of 
Progres-  quantities  which  increase  or  diminish  constantly  by 
the  same  amount.  But  the  theory  of  geometrical  pro- 
gressions is  more  difficult  and  more  important,  as  a 
large  number  of  interesting  questions  depend  upon  it 
— for  example,  all  problems  of  compound  interest,  all 
problems  that  relate  to  discount,  and  many  others  of 
like  nature. 

In  general,  quantities  in  geometrical  proportion 
are  produced,  when  a  quantity  increases  and  the  force 
generating  the  increase,  so  to  speak,  is  proportional 
to  that  quantity.  It  has  been  observed  that  in  coun- 
tries where  the  means  of  subsistence  are  easy  of  ac- 
quisition, as  in  the  first  American  colonies,  the  popu- 
lation is  doubled  at  the  expiration  of  twenty  years  ;  if 
it  is  doubled  at  the  end  of  twenty  years  it  will  be  quad- 
rupled at  the  end  of  forty,  octupled  at  the  end  of  sixty, 
and  so  on  ;  the  result  being,  as  we  see,  a  geometrical 
progression,  corresponding  to  intervals  of  time  in 
arithmetical  progression.  It  is  the  same  with  com- 
pound interest.  If  a  given  sum  of  money  produces, 
at  the  expiration  of  a  certain  time,  a  certain  sum,  at 
the  end  of  double  that  time,  the  original  sum  will  have 
produced  an  equivalent  additional  sum,  and  in  addi- 
tion the  sum  produced  in  the  first  space  of  time  will, 
in  its  proportion,  likewise  have  produced  during  the 
second  space  of  time  a  certain  sum  ;  and  so  with  the 
rest.  The  original  sum  is  commonly  called  the  prin- 


ON  ARITHMETIC.  15 

cipal,  the  sum  produced  the  interest,  and  the  constant 
ratio  of  the  principal  to  the  interest  per  annum,  the 
rate.  Thus,  the  rate  twenty  signifies  that  the  interest  Compound 

interest. 

is  the  twentieth  part  of  the  principal, — a  rate  which 
is  commonly  called  5 per  cent.,  5  being  the  twentieth 
part  of  100.  On  this  basis,  the  principal,  at  the  end 
of  one  year,  will  have  increased  by  its  one-twentieth 
part ;  consequently,  it  will  have  been  augmented  in 
the  ratio  of  21  to  20.  At  the  end  of  two  years,  it  will 
have  been  increased  again  in  the  same  ratio,  that  is  in 
the  ratio  of  f  i-  multiplied  by  \\ ;  at  the  end  of  three 
years,  in  the  ratio  of  f  1  multiplied  twice  by  itself  ;  and 
so  on.  In  this  manner  we  shall  find  that  at  the  end  of 
fifteen  years  it  will  almost  have  doubled  itself,  and  that 
at  the  end  of  fifty-three  years  it  will  have  increased 
tenfold.  Conversely,  then,  since  a  sum  paid  now  will 
be  doubled  at  the  end  of  fifteen  years,  it  is  clear  that 
a  sum  not  payable  till  after  the  expiration  of  fifteen 
years  is  now  worth  only  one-half  its  amount.  This 
is  what  is  termed  the  present  value  of  a  sum  payable 
at  the  end  of  a  certain  time  ;  and  it  is  plain,  that  to 
find  that  value,  it  is  only  necessary  to  divide  the  sum 
promised  by  the  fraction  £i,  or  to  multiply  it  by  the 
fraction  f £,  as  many  times  as  there  are  years  for  the 
sum  to  run.  In  this  way  we  shall  find  that  a  sum 
payable  at  the  end  of  fifty-three  years,  is  worth  at 
present  only  one-tenth.  From  this  it  is  evident  what 
little  advantage  is  to  be  derived  from  surrendering  the 
absolute  ownership  of  a  sum  of  money  in  order  to  ob- 


l6  ON  ARITHMETIC. 

tain   the  enjoyment   of  it   for  a  period   of  only  fifty 
years,  say;  seeing  that  we  gain  by  such  a  transaction 

Present        only  one-tenth  in  actual  use,  whilst  we  lose  the  owner- 
values  and         . 
annuities,     ship  of  the  property  forever. 

In  annuities,  the  consideration  of  interest  is  com- 
bined with  that  of  the  probability  of  life;  and  as 
every  one  is  prone  to  believe  that  he  will  live  very 
long,  and  as,  on  the  other  hand,  one  is  apt  to  under- 
estimate the  value  of  property  which  must  be  aban- 
doned on  death,  a  peculiar  temptation  arises,  when 
one  is  without  children,  to  invest  one's  fortune,  wholly 
or  in  part,  in  annuities.  Nevertheless,  when  put  to 
the  test  of  rigorous  calculation,  annuities  are  not 
found  to  offer  sufficient  advantages  to  induce  people 
to  sacrifice  for  them  the  ownership  of  the  original 
capital.  Accordingly,  whenever  it  has  been  attempted 
to  create  annuities  sufficiently  attractive  to  induce  in- 
dividuals to  invest  in  them,  it  has  been  necessary  to 
offer  them  on  terms  which  are  onerous  to  the  com- 
pany. 

But  we  shall  have  more  to  say  on  this  subject  when 
we  expound  the  theory  of  annuities,  which  is  a  branch 
of  the  calculus  of  probabilities. 

I  shall  conclude  the  present  lecture  with  a  word 
on  logarithms.  The  simplest  idea  which  we  can  form 
of  the  theory  of  logarithms,  as  they  are  found  in  the 
ordinary  tables,  is  that  of  conceiving  all  numbers 
as  powers  of  10;  the  exponents  of  these  powers., 
then,  will  be  the  logarithms  of  the  numbers.  From 


ON  ARITHMETIC.  1 7 

this  it  is  evident  that  the  multiplication  and  division 
of  two  numbers  is  reducible  to  the  addition  and  sub- 
traction of  their  respective  exponents,  that  is,  of  their  Logarithms 
logarithms.  And,  consequently,  involution  and  the 
extraction  of  roots  are  reducible  to  multiplication  and 
division,  which  is  of  immense  advantage  in  arithmetic 
and  renders  logarithms  of  priceless  value  in  that  sci- 
ence. 

But  in  the  period  when  logarithms  were  invented, 
mathematicians  were  not  in  possession  of  the  theory 
of  powers.  They  did  not  know  that  the  root  of  a  num- 
ber could  be  represented  by  a  fractional  power.  The 
following  was  the  way  in  which  they  approached  the 
problem. 

The  primitive  idea  was  that  of  two  corresponding 
progressions,  one  arithmetical,  and  the  other  geomet- 
rical. In  this  way  the  general  notion  of  a  logarithm 
was  reached.  But  the  means  for  rinding  the  loga- 
rithms of  all  numbers  were  still  lacking.  As  the  num- 
bers follow  one  another  in  arithmetical  progression,  it 
was  requisite,  in  order  that  they  might  all  be  found 
among  the  terms  of  a  geometrical  progression,  so  to 
establish  that  progression  that  its  successive  terms 
should  differ  by  extremely  small  quantities  from  one 
another ;  and,  to  prove  the  possibility  of  expressing 
all  numbers  in  this  way,  Napier,  the  inventor,  first 
considered  them  as  expressed  by  lines  and  parts  of 
lines,  and  these  lines  he  considered  as  generated  by 


1 8  ON  ARITHMETIC. 

the  continuous  motion  of  a  point,  which  was  quite 
natural. 

Napier  He  considered,  accordingly,  two  lines,  the  first  of 

which  was  generated  by  the  motion  of  a  point  describ- 
ing in  equal  times  spaces  in  geometrical  progression, 
and  the  other  generated  by  a  point  which  described 
spaces  that  increased  as  the  times  and  consequently 
formed  an  arithmetical  progression  corresponding  to 
the  geometrical  progression.  And  he  supposed,  for 
the  sake  of  simplicity,  that  the  initial  velocities  of 
these  two  points  were  equal.  This  gave  him  the  loga- 
rithms, at  first  called  natural,  and  afterwards  hyper- 
bolical, when  it  was  discovered  that  they  could  be  ex- 
pressed as  parts  of  the  area  included  between  a 
hyperbola  and  its  asymptotes.  By  this  method  it  is 
clear  that  to  find  the  logarithm  of  any  given  number, 
it  is  only  necessary  to  take  a  part  on  the  first  line 
equal  to  the  given  number,  and  to  seek  the  part  on 
the  second  line  which  shall  have  been  described  in 
the  same  interval  of  time  as  the  part  on  the  first. 

Conformably  to  this  idea,  if  we  take  as  the  two 
first  terms  of  our  geometrical  progression  the  numbers 
with  very  small  differences  1  and  1.0000001,  and  as 
those  of  our  arithmetical  progression  0  and  0.0000001, 
and  if  we  seek  successively,  by  the  known  rules,  all 
the  following  terms  of  the  two  progressions,  we  shall 
find  that  the  number  2  expressed  approximately  to  the 
eighth  place  of  decimals  is  the  6931472th  term  of  thf- 
geometrical  progression,  that  is,  that  the  logarithm  ov 


ON  ARITHMETIC.  IQ 

2  is  0.6931472.  The  number  10  will  be  found  to  be  the 
23025851th  term  of  the  same  progression;  therefore, 
the  logarithm  of  10  is  2.3025851,  and  so  with  the  rest,  origin  of 

logarithms 

But  Napier,  having  to  determine  only  the  logarithms 
of  numbers  less  than  unity  for  the  purposes  of  trigo- 
nometry, where  the  sines  and  cosines  of  angles  are 
expressed  as  fractions  of  the  radius,  considered  a  de- 
creasing geometrical  progression  of  which  the  first 
two  terms  were  1  and  0.9999999;  and  of  this  progres- 
sion he  determined  the  succeeding  terms  by  enormous 
computations.  On  this  last  hypothesis,  the  logarithm 
which  we  have  just  found  for  2  becomes  that  of  the 
number  £  or  0.5,  and  that  of  the  number  10  becomes 
that  of  the  number  T^  or  0.1  ;  as  is  readily  apparent 
from  the  nature  of  the  two  progressions. 

Napier's  work  appeared  in  1614.  Its  utility  was 
felt  at  once.  But  it  was  also  immediately  seen  that  it 
would  conform  better  to  the  decimal  system  of  our 
arithmetic,  and  would  be  simpler,  if  the  logarithm  of 
10  were  made  unity,  conformably  to  which  that  of  100 
would  be  2,  and  so  with  the  rest.  To  that  end,  in- 
stead of  taking  as  the  first  two  terms  of  our  geometrical 
progression  the  numbers  1  and  0.0000001,  we  should 
have  to  take  the  numbers  1  and  1.0000002302,  retain- 
ing 0  and  0.0000001  as  the  corresponding  terms  of  the 
arithmetical  progression.  Whence  it  will  be  seen, 
that,  while  the  point  which  is  supposed  to  generate  by 
its  motion  the  geometrical  line,  or  the  numbers,  is 
describing  the  very  small  portion  0.0000002302  .  .  .  , 


2O  ON  ARITHMETIC. 

the  other  point,  the  office  of  which  is  to  generate 
simultaneously  the  arithmetical  line,  will  have  de- 
Briggs  scribed  the  portion  0.0000001  ;  and  that  therefore  the 
viacq.  spaces  described  in  the  same  time  by  the  two  points 
at  the  beginning  of  their  motion,  that  is  to  say,  their 
initial  velocities,  instead  of  being  equal,  as  in  the 
preceding  system,  will  be  in  the  proportion  of  the 
numbers  2.302  ...  to  1,  where  it  will  be  remarked 
that  the  number  2.302  ...  is  exactly  the  number 
which  in  the  original  system  of  natural  logarithms 
stood  for  the  logarithm  of  10, — a  result  demonstrable 
a  priori,  as  we  shall  see  when  we  come  to  apply 
the  formulae  of  algebra  to  the  theory  of  logarithms. 
Briggs,  a  contemporary  of  Napier,  is  the  author  of  this 
change  in  the  system  of  logarithms,  as  he  is  also  of 
the  tables  of  logarithms  now  in  common  use.  A  por- 
tion of  these  was  calculated  by  Briggs  himself,  and 
the  remainder  by  Vlacq,  a  Dutchman. 

These  tables  appeared  at  Gouda,  in  1628.  They 
contain  the  logarithms  of  all  numbers  from  1  to  100000 
to  ten  decimal  places,  and  are  now  extremely  rare. 
But  it  was  afterwards  discovered  that  for  ordinary  pur- 
poses seven  decimals  were  sufficient,  and  the  loga- 
rithms are  found  in  this  form  in  the  tables  which  are 
used  to-day.  Briggs  and  Vlacq  employed  a  number 
of  highly  ingenious  artifices  for  facilitating  their  work. 
The  device  which  offered  itself  most  naturally  and 
which  is  still  one  of  the  simplest,  consists  in  taking 
the  numbers  1,  10,  100,  .  .  .  ,  of  which  the  logarithms 


ON  ARITHMETIC.  21 

are  0,  1,  2,  .  .  .  ,  and  in  interpolating  between  the  suc- 
cessive terms  of  these  two  series  as  many  correspond- 
ing  terms   as  we   desire,   in  the  first  series  by  geo-  Computa- 
tion of  log- 
metrical   mean   proportionals  and   in  the  second   by  arithms. 

arithmetical  means.  In  this  manner,  when  we  have 
arrived  at  a  term  of  the  first  series  approaching,  to  the 
eighth  decimal  place,  the  number  whose  logarithm 
we  seek,  the  corresponding  term  of  the  other  series 
will  be,  to  the  eighth  decimal  place  approximately, 
the  logarithm  of  that  number.  Thus,  to  obtain  the 
logarithm  of  2,  since  2  lies  between  1  and  10,  we  seek 
first  by  the  extraction  of  the  square  root  of  10,  the 
geometrical  mean  between  1  and  10,  which  we  find  to 
be  3.16227766,  while  the  corresponding  arithmetical 
mean  between  0  and  1  is  ^  or  0.50000000;  we  are 
assured  thus  that  this  last  number  is  the  logarithm  of 
the  first.  Again,  as  2  lies  between  1  and  3.16227766, 
the  number  just  found,  we  seek  in  the  same  manner 
the  geometrical  mean  between  these  two  numbers, 
and  find  the  number  1.77827941.  As  before,  taking 
the  arithmetical  mean  between  0  and  5.0000000,  we 
shall  have  for  the  logarithm  of  1.77827941  the  num- 
ber 0.25000000.  Again,  2  lying  between  1.77827941 
and  3.16227766,  it  will  be  necessary,  for  still  further 
approximation,  to  find  the  geometrical  mean  between 
these  two,  and  likewise  the  arithmetical  mean  be- 
tween their  logarithms.  And  so  on.  In  this  manner, 
by  a  large  number  of  similar  operations,  we  find  that 
the  logarithm  of  2  is  0.3010300,  that  of  3  is  0.4771213, 


22  ON  ARITHMETIC. 

and  so  on,  not  carrying  the  degree  of  exactness  be- 
yond the  seventh  decimal  place.      But  the  preceding 
Value  of      calculation  is  necessary  only  for  prime  numbers  ;  be- 

the  history  1-1  i 

of  science,  cause  the  logarithms  of  numbers  which  are  the  pro- 
duct of  two  or  several  others,  are  found  by  simply 
taking  the  sum  of  the  logarithms  of  their  factors. 

As  for  the  rest,  since  the  calculation  of  logarithms 
is  now  a  thing  of  the  past,  except  in  isolated  instances, 
it  may  be  thought  that  the  details  into  which  we  have 
here  entered  are  devoid  of  value.  We  may,  however, 
justly  be  curious  to  know  the  trying  and  tortuous 
paths  which  the  great  inventors  have  trodden,  the  dif- 
ferent steps  which  they  have  taken  to  attain  their  goal, 
and  the  extent  to  which  we  are  indebted  to  these  ver- 
itable benefactors  of  the  human  race.  Such  knowl- 
edge, moreover,  is  not  matter  of  idle  curiosity.  It  can 
afford  us  guidance  in  similar  inquiries  and  sheds  an 
increased  light  on  the  subjects  with  which  we  are 
employed. 

Logarithms  are  an  instrument  universally  employed 
in  the  sciences,  and  in  the  arts  depending  on  calcula- 
tion. The  following,  for  example,  is  a  very  evident 
application  of  their  use. 

Persons  not  entirely  unacquainted  with  music  know 
that  the  different  notes  of  the  octave  are  expressed  by 
numbers  which  give  the  divisions  of  a  stretched  cord 
producing  those  notes.  Thus,  the  principal  note  be- 
ing denoted  by  1,  its  octave  will  be  denoted  by  £, 
its  fifth  by  f ,  its  third  by  4,  its  fourth  by  f ,  its  second 


ON  ARITHMETIC.  23 

by  f,  and  so  on.  The  distance  of  one  of  these  notes 
from  that  next  adjacent  to  it  is  called  an  interval,  and 
is  measured,  not  by  the  difference,  but  by  the  ratio  of 
the  numbers  expressing  the  two  sounds.  Thus,  the 
interval  between  the  fourth  and  fifth,  which  is  called 
the  major  tone,  is  regarded  as  sensibly  double  of  that 
between  the  third  and  fourth,  which  is  called  the  semi- 
major.  In  fact,  the  first  being  expressed  by  f,  the 
second  by  \\ ,  it  can  be  easily  proved  that  the  first 
does  not  differ  by  much  from  the  square  of  the  second. 
Now,  it  is  clear  that  this  conception  of  intervals,  on  Musical 
which  the  whole  theory  of  temperament  is  founded,  ^^W 
conducts  us  naturally  to  logarithms.  For  if  we  ex- 
press the  value  of  the  different  notes  by  the  loga- 
rithms of  the  lengths  of  the  cords  answering  to  them, 
then  the  interval  of  one  note  from  another  will  be 
expressed  by  the  simple  difference  of  values  of  the 
two  notes  ;  and  if  it  were  required  to  divide  the  octave 
into  twelve  equal  semi-tones,  which  would  give  the 
temperament  that  is  simplest  and  most  exact,  we 
should  simply  have  to  divide  the  logarithm  of  one 
half,  the  value  of  the  octave,  into  twelve  equal  parts. 


AN 


LECTURE  II. 

ON  THE  OPERATIONS  OF  ARITHMETIC. 

,N  ANCIENT  writer  once  remarked  that  arith- 
metic and  geometry  were  the  wings  of  matkemat- 
Arithmetic  ics.  I  believe  we  can  say,  without  metaphor,  that 
these  two  sciences  are  the  foundation  and  essence  of 
all  the  sciences  that  treat  of  magnitude.  But  not 
only  are  they  the  foundation,  they  are  also,  so  to 
speak,  the  capstone  of  these  sciences.  For,  whenever 
we  have  reached  a  result,  in  order  to  make  use  of  it, 
it  is  requisite  that  it  be  translated  into  numbers  or 
into  lines  ;  to  translate  it  into  numbers,  arithmetic  is 
necessary ;  to  translate  it  into  lines,  we  must  have 
recourse  to  geometry. 

The  importance  of  arithmetic,  accordingly,  leads 
me  to  the  further  discussion  of  that  subject  to-day, 
although  we  have  begun  algebra.  I  shall  take  up  its 
several  parts,  and  shall  offer  new  observations,  which 
will  serve  to  supplement  what  I  have  already  ex- 
pounded to  you.  I  shall  employ,  moreover,  the  geo- 
metrical calculus,  wherever  that  is  necessary  for  giv- 


ON  THE  OPERATIONS  OF  ARITHMETIC.  25 

ing    greater    generality    to    the    demonstrations    and 
methods. 

First,  then,  as  regards  addition,  there  is  nothing 
to  be  added  to  what  has  already  been  said.  Addition 
is  an  operation  so  simple  in  character  that  its  concep- 
tion is  a  matter  of  course.  But  with  regard  to  sub-  New 

method  of 

traction,  there  is  another  manner  of  performing  that  subtraction 
operation  which  is  frequently  more  advantageous  than 
the  common  method,  particularly  for  those  familiar 
with  it.  It  consists  in  converting  the  subtraction  into 
addition  by  taking  the  complement  of  every  figure  of 
the  number  which  is  to  be  subtracted,  first  with  re- 
spect to  10  and  afterwards  with  respect  to  9.  Sup- 
pose, for  example,  that  the  number  2635  is  to  be  sub- 
tracted from  the  number  7853.  Instead  of  saying  5 

7853 
2635 


5218 

from  13  leaves  8  ;  3  from  4  leaves  1  ;  6  from  8  leaves 
2  ;  and  2  from  7  leaves  5,  giving  a  total  remainder  of 
5218, —  I  say  :  5  the  complement  of  5  with  respect  to 
10  added  to  3  gives  8, — I  write  down  8;  6  the  com- 
plement of  3  with  respect  to  9  added  to  5  gives  11, — 
I  write  down  1  and  carry  1  ;  3  the  complement  of  6 
with  respect  to  9,  plus  9,  by  reason  of  the  1  carried, 
gives  12, — I  put  down  2  and  carry  1  ;  lastly,  7  the 
complement  of  2  with  respect  to  9  plus  8,  on  account 
of  the  1  carried,  gives  15, — I  put  down  5  and  this  time 
carry  nothing,  for  the  operation  is  completed,  and  the 


26  ON  THE  OPERATIONS  OF  ARITHMETIC. 

last  10  which  was  borrowed  in  the  course  of  the  oper- 
ation must  be  rejected.  In  this  manner  we  obtain  the 
same  remainder  as  above,  5218. 

The  foregoing  method  is  extremely  convenient 
Subtraction  when  the  numbers  are  large ;  for  in  the  common 
method  of  subtraction,  where  borrowing  is  necessary 
in  subtracting  single  numbers  from  one  another,  mis- 
takes are  frequently  made,  whereas  in  the  method 
with  which  we  are  here  concerned  we  never  borrow 
but  simply  carry,  the  subtraction  being  converted  into 
addition.  With  regard  to  the  complements  they  are 
discoverable  at  the  merest  glance,  for  every  one  knows 
that  3  is  the  complement  of  7  with  respect  to  10,  4 
the  complement  of  5  with  respect  to  9,  etc.  And  as 
to  the  reason  of  the  method,  it  too  is  quite  palpable. 
The  different  complements  taken  together  form  the 
total  complement  of  the  number  to  be  subtracted 
either  with  respect  to  10  or  100  or  1000,  etc.,  accord- 
ing as  the  number  has  1,  2,  3  ...  figures ;  so  that  the 
operation  performed  is  virtually  equivalent  to  first 
adding  10,  100,  1000  ...  to  the  minuend  and  then 
taking  the  subtrahend  from  the  minuend  as  so  aug- 
mented. Whence  it  is  likewise  apparent  why  the  10 
of  the  sum  found  by  the  last  partial  addition  must  be 
rejected. 

As  to  multiplication,  there  are  various  abridged 
methods  possible,  based  on  the  decimal  system  of 
numbers.  In  multiplying  by  10,  for  example,  we  have, 
as  we  know,  simply  to  add  a  cipher;  in  multiplying 


ON  THE  OPERATIONS  OF  ARITHMETIC.  27 

by  100  we  add  two  ciphers;   by  1000,  three  ciphers, 

etc.    Consequently,  to  multiply  by  any  aliquot  part  of 

10,  for  example  5,  we  have  simply  to  multiply  by  10  Abridged 

and  then  divide  by  2  ;  to  multiply  by  25  we  multiply  tion. 

by  100  and  divide  by  4,  and  so  on  for  all  the  products 

of  5. 

When  decimal  numbers  are  to  be  multiplied  by 
decimal  numbers,  the  general  rule  is  to  consider  the 
two  numbers  as  integers  and  when  the  operation  is 
finished  to  mark  off  from  the  right  to  the  left  as  many 
places  in  the  product  as  there  are  decimal  places  in 
the  multiplier  and  the  multiplicand  together.  But  in 
practice  this  rule  is  frequently  attended  with  the  in- 
convenience of  unnecessarily  lengthening  the  opera- 
tion, for  when  we  have  numbers  containing  decimals 
these  numbers  are  ordinarily  exact  only  to  a  certain 
number  of  places,  so  that  it  is  necessary  to  retain  in 
the  product  only  the  decimal  places  of  an  equivalent 
order.  For  example,  if  the  multiplicand  and  the  multi- 
plier each  contain  two  places  of  decimals  and  are  ex- 
act only  to  two  decimal  places,  we  should  have  in  the 
product  by  the  ordinary  method  four  decimal  places, 
the  two  last  of  which  we  should  have  to  reject  as  use- 
less and  inexact.  I  shall  give  you  now  a  method  for 
obtaining  in  the  product  only  just  so  many  decimal 
places  as  you  desire. 

I  observe  first  that  in  the  ordinary  method  of'mul- 
tiplying  we  begin  with  the  units  of  the  multiplier  which 
we  multiply  with  the  units  of  the  multiplicand,  and  so 


28  ON  THE  OPERATIONS  OF  ARITHMETIC. 

continue  from  the  right  to  the  left.   But  there  is  noth- 
ing compelling  us  to  begin  at  the  right  of  the  multi- 
inverted      plier.     We  may  equally  well  begin  at  the  left.     And 

inultiplica-  '     . 

tion.  I  cannot  in  truth  understand  why  the  latter  method 

should  not  be  preferred,  since  it  possesses  the  advan- 
tage of  giving  at  once  the  figures  having  the  greatest 
value,  and  since,  in  the  majority  of  cases  where  large 
numbers  are  multiplied  together,  it  is  just  these  last 
and  highest  places  that  concern  us  most  ;  we  fre- 
quently, in  fact,  perform  multiplications  only  to  find 
what  these  last  figures  are.  And  herein,  be  it  par- 
enthetically remarked,  consists  one  of  the  great  ad- 
vantages in  calculating  by  logarithms,  which  always 
give,  be  it  in  multiplication  or  division,  in  involution 
or  evolution,  the  figures  in  the  descending  order  of 
their  value,  beginning  with  the  highest  and  proceed- 
ing from  the  left  to  the  right. 

By  performing  multiplication  in  this  manner,  no 
difference  is  caused  in  the  total  product.  The  sole 
distinction  is,  that  by  the  new  method  the  first  line, 
the  first  partial  product,  is  that  which  in  the  ordinary 
method  is  last,  and  the  second  partial  product  is  that 
which  in  the  ordinary  method  is  next  to  the  last,  and 
so  with  the  rest. 

Where  whole  numbers  are  concerned  and  the  exact 
product  is  required,  it  is  indifferent  which  method  we 
employ.  But  when  decimal  places  are  involved  the 
prime  essential  is  to  have  the  figures  of  the  whole 
numbers  first  in  the  product  and  to  descend  after- 


ON  THE  OPERATIONS  OF  ARITHMETIC.  2Q 

wards  successively  to  the  figures  of  the  decimal  parts, 
instead  of,  as  in  the  ordinary  method,  beginning  with 
the  last  decimal  places  and  successively  ascending  to 
the  figures  forming  the  whole  numbers. 

In  applying  this  method  practically,  we  write  the 
multiplier  underneath  the  multiplicand  so  that  the 
units'  figure  of  the  multiplier  falls  beneath  the  last 
figure  of  the  multiplicand.  We  then  begin  with  the 
last  left-hand  figure  of  the  multiplier  which  we  multi- 
ply  as  in  the  ordinary  method  by  all  the  figures  of  the 
multiplicand,  beginning  with  the  last  to  the  right  and 
proceeding  successively  to  the  left ;  observing  that  the 
first  figure  of  the  product  is  to  be  placed  underneath 
the  figure  with  which  we  are  multiplying,  while  the 
others  follow  in  their  successive  order  to  the  left.  We 
proceed  in  the  same  manner  with  the  second  figure  of 
the  multiplier,  likewise  placing  beneath  this  figure  the 
first  figure  of  the  product,  and  so  on  with  the  rest. 
The  place  of  the  decimal  point  in  these  different  pro- 
ducts will  be  the  same  as  in  the  multiplicand,  that  is 
to  say,  the  units  of  the  products  will  all  fall  in  the 
same  vertical  line  with  those  of  the  multiplicand  and 
consequently  those  of  the  sum  of  all  the  products  or 
of  the  total  product  will  also  fall  in  that  line.  In  this 
manner  it  is  an  easy  matter  to  calculate  only  as  many 
decimal  places  as  we  wish.  I  give  below  an  example 
of  this  method  in  which  the  multiplicand  is  437.25 
and  the  multiplier  27.34  : 


30  ON  THE  OPERATIONS  OF  ARITHMETIC. 

437.25 
27.34 


The  new 
method  ex- 
emplified 


8745 

0 

3060 

75 

131 

17  5 

17 

49  00 

11954|41  50 

I  have  written  all  the  decimals  in  the  product,  but 
it  is  easy  to  see  how  we  may  omit  calculating  the  deci- 
mals which  we  wish  to  neglect.  The  vertical  line  is 
used  to  mark  more  distinctly  the  place  of  the  decimal 
point. 

The  preceding  rule  appears  to  me  simpler  and 
more  natural  than  that  which  is  attributed  to  Oughtred 
and  which  consists  in  writing  the  multiplier  under- 
neath the  multiplicand  in  the  reverse  order. 

There  is  one  more  point,  finally,  to  be  remarked 
in  connexion  with  the  multiplication  of  numbers  con- 
taining decimals,  and  that  is  that  we  may  alter  the 
place  of  the  decimal  point  of  either  number  at  will. 
For  seeing  that  moving  the  decimal  point  from  the 
right  to  the  left  in  one  of  the  numbers  is  equivalent  to 
dividing  the  number  by  10,  by  100,  or  by  1000...,  and 
that  moving  the  decimal  point  back  in  the  other  num- 
ber the  same  number  of  places  from  the  left  to  the 
right  is  tantamount  to  multiplying  that  number  by  10, 
100,  or  1000,  .  .  .  ,  it  follows  that  we  may  push  the 
decimal  point  forward  in  one  of  the  numbers  as  many 
places  as  we  please  provided  we  move  it  back  in  the 
other  number  the  same  number  of  places,  without  in 


ON  THE  OPERATIONS  OF  ARITHMETIC.  31 

any  wise  altering  the  product.  In  this  way  we  can 
always  so  arrange  it  that  one  of  the  two  numbers  shall 
contain  no  decimals — which  simplifies  the  question. 

Division  is  susceptible  of  a  like  simplification,  for 
since  the  quotient  is  not  altered  by  multiplying  or  di-  Division  of 
viding  the  dividend  and  the  divisor  by  the  same  num- 
ber, it  follows  that  in  division  we  may  move  the  deci- 
mal point  of  both  numbers  forwards  or  backwards  as 
many  places  as  we  please,  provided  we  move  it  the 
same  distance  in  each  case.  Consequently,  we  can 
always  reduce  the  divisor  to  a  whole  number — which 
facilitates  infinitely  the  operation  for  the  reason  that 
when  there  are  decimal  places  in  the  dividend  only, 
we  may  proceed  with  the  division  by  the  common 
method  and  neglect  all  places  giving  decimals  of  a 
lower  order  than  those  we  desire  to  take  account  of. 

You  know  the  remarkable  property  of  the  number 
9,  whereby  if  a  number  be  divisible  by  9  the  sum  of 
its  digits  is  also  divisible  by  9.  This  property  enables 
us  to  tell  at  once,  not  only  whether  a  number  is  divis- 
ible by  9  but  also  what  is  its  remainder  from  such  di- 
vision. For  we  have  only  to  take  the  sum  of  its  digits 
and  to  divide  that  sum  by  9,  when  the  remainder  will 
be  the  same  as  that  of  the  original  number  divided 
by  9. 

The  demonstration  of  the  foregoing  proposition  is 
not  difficult.  It  reposes  upon  the  fact  that  the  num- 
bers 10  less  1,  100  less  1,  1000  less  1,  ...  are  all  di- 


32  ON  THE  OPERATIONS  OF  ARITHMETIC. 

visible  by  9,  —  which  seeing  that  the  resulting  numbers 
are  9,  99,  999,  ...  is  quite  obvious. 

If,   now,  you  subtract  from  a  given   number  the 

sum  of  all  its  digits,  you  will  have  as  your  remainder 

Property      the  tens'  digit  multiplied  by  9,   the  hundreds'  digit 

of  the  num- 

ber 9.  multiplied  by  99,  the  thousands'  digit  multiplied  by 
999,  and  so  on,  —  a  remainder  which  is  plainly  divis- 
ible by  9.  Consequently,  if  the  sum  of  the  digits  is 
divisible  by  9,  the  original  number  itself  will  be  so 
divisible,  and  if  it  is  not  divisible  by  9  the  original 
number  likewise  will  not  be  divisible  thereby.  But 
the  remainder  in  the  one  case  will  be  the  same  as  in 
the  other. 

In  the  case  of  the  number  9,  it  is  evident  imme- 
diately that  10  less  1,  100  less  1,  ...  are  divisible  by 
9  ;  but  algebra  demonstrates  that  the  property  in 
question  holds  good  for  every  number  a.  For  it  can 
be  shown  that 

a  —  I,  a*  —  l,  a*  —  l,  a4  —1,  .  .  . 

are  all  quantities  divisible  by  a  —  1,   actual  division 
giving  the  quotients 


The  conclusion  is  therefore  obvious  that  the  afore- 
said property  of  the  number  9  holds  good  in  our  de- 
cimal system  of  arithmetic  because  9  is  10  less  1,  and 
that  in  any  other  system  founded  upon  the  progres- 
sion a,  a2,  as,  .  .  .  .  the  number  a  —  1  would  enjoy  the 
same  property.  Thus  in  the  duodecimal  system  it 


ON  THE  OPERATIONS  OF  ARITHMETIC.  33 

would  be  the  number  11  ;  and  in  this  system  every 
number,  the  sum  of  whose  digits  was  divisible  by  11, 
would  also  itself  be  divisible  by  that  number. 

The  foregoing  property  of  the  number  9,  now,  ad- 
mits of  generalisation,  as  the  following  consideration  Property  of 

....  „.  ,  the  number 

will  show,  bmce  every  number  in  our  system  is  rep-  9  general- 
resented  by  the  sum  of  certain  terms  of  the  progres  ' 
sion  1,  10,  100,  1000,  .  .  .  ,  each  multiplied  by  one  of 
the  nine  digits  1,  2,  3,  4,  ....  9,  it  is  easy  to  see  that 
the  remainder  resulting  from  the  division  of  any  num- 
ber by  a  given  divisor  will  be  equal  to  the  sum  of  the 
remainders  resulting  from  the  division  of  the  terms  1, 
10,  100,  1000,  ...  by  that  divisor,  each  multiplied  by 
the  digit  showing  how  many  times  the  corresponding 
term  has  been  taken.  Hence,  generally,  if  the  given 
divisor  be  called  D,  and  if  m,  n,  p,  .  .  .  be  the  remain- 
ders of  the  division  of  the  numbers  1,  10,  100,  1000 
by  D,  the  remainder  from  the  division  of  any  number 
whatever  N,  of  which  the  characters  proceeding  from 
the  right  to  the  left  are  a,  b,  c,  .  .  .  ,  by  D  will  obviously 
be  equal  to 

ma  -f-  nb  -\- pc  -\-  .  .  .  . 

Accordingly,  if  for  a  given  divisor  D  we  know  the  re- 
mainders m,  «,/,...,  which  depend  solely  upon  that 
divisor  and  which  are  always  the  same  for  the  same 
divisor,  we  have  only  to  write  the  remainders  under- 
neath the  original  number,  proceeding  from  the  right 
to  the  left,  and  then  to  find  the  different  products  of 


34  ON  THE  OPERATIONS  OF  ARITHMETIC. 

each  digit  of  the  number  by  the  digit  which  is  under- 
neath it.  The  sum  of  all  these  products  will  be  the 

Theory  of     total  remainder  resulting  from  the  division  of  the  pro- 
remainders 

posed  number  by  the  same  divisor  D.   And  if  the  sum 

found  is  greater  than  Z>,  we  can  proceed  in  the  same 
manner  to  seek  its  remainder  from  division  by  D,  and 
so  on  until  we  arrive  finally  at  a  remainder  which  is 
less  than  D,  which  will  be  the  true  remainder  sought. 
It  follows  from  this  that  the  proposed  number  cannot 
be  exactly  divisible  by  the  given  divisor  unless  the 
last  remainder  found  by  this  method  is  zero. 

The  remainders  resulting  from  the  division  of  the 
terms  1,  10,  100,  ....  1000,  by  9  are  always  unity. 
Hence,  the  sum  of  the  digits  of  any  number  whatever 
is  the  remainder  resulting  from  the  division  of  that 
number  by  9.  The  remainders  resulting  from  the  di- 
vision of  the  same  terms  by  8  are  1,  2,  4,  0,  0,  0,  .... 
We  shall  obtain,  accordingly,  the  remainder  resulting 
from  dividing  any  number  by  8,  by  taking  the  sum 
of  the  first  digit  to  the  right,  the  second  digit  next 
thereto  to  the  left  multiplied  by  2,  and  the  third  digit 
multiplied  by  4. 

The  remainders  resulting  from  the  divisions  of  the 
terms  1,  10,  100,  1000,  ...  by  7  are  1,  3,  2,  G,  4,  5, 
1,  3,  .  .  .  ,  where  the  same  remainders  continually  re- 
cur in  the  same  order.  If  I  have,  now,  the  number 
13527541  to  be  divided  by  7,  I  write  it  thus  with  the 
above  remainders  underneath  it : 


ON  THE  OPERATIONS  OF  ARITHMETIC.  35 

13527541 
:•!]  546231 


Test  of 

i£i  divisibility 

10  by  7. 

42 

8 
25 

3 

3 


104 
231 

4 

0 

_2 

~6 


Taking  the  partial  products  and  adding  them,  I 
obtain  104,  which  would  be  the  remainder  from  the 
division  of  the  given  number  by  7,  were  it  not  greater 
than  the  divisor.  I  accordingly  repeat  the  operation 
with  this  remainder,  and  find  for  my  second  remainder 
6,  which  is  the  real  remainder  in  question. 

I  have  still  to  remark  with  regard  to  the  preceding 
remainders  and  the  multiplications  which  result  from 
them,  that  they  may  be  simplified  by  introducing  nega- 
tive remainders  in  the  place  of  remainders  which  are 
greater  than  half  the  divisor,  and  to  accomplish  this 
we  have  simply  to  subtract  the  divisor  from  each  of 
such  remainders.  We  obtain  thus,  instead  of  the  re- 
mainders 6,  5,  4,  the  following : 


36  ON  THE  OPERATIONS  OF  ARITHMETIC. 

The  remainders  for  the  divisor  7,  accordingly,  are 

1      Q     9  1  Q  9     1      q 

X>     °5     ">     -"-5     °>     ^>     -I,     «J,     •     .     • 

and  so  on  to  infinity. 
Negative  The  preceding  example,  then,  takes  the  following 

remainders 

form  : 

13527541 
31231231 


7     1 

6  12 

U)  10 

~2~3     3 
3 

29 
subtract  23 

6 

I  have  placed  a  bar  beneath  the  digits  which  are 
to  be  taken  negatively,  and  I  have  subtracted  the  sum 
of  the  products  of  these  numbers  by  those  above  them 
from  the  sum  of  the  other  products. 

The  whole  question,  therefore,  resolves  itself  into 
finding  for  every  divisor  the  remainders  resulting  from 
dividing  1,  10,  100,  1000  by  that  divisor.  This  can  be 
readily  done  by  actual  division ;  but  it  can  be  accom- 
plished more  simply  by  the  following  consideration. 
If  r  be  the  remainder  from  the  division  of  10  by  a 
given  divisor,  r1  .will  be  the  remainder  from  the  divi- 
sion of  100,  the  square  of  10,  by  that  divisor;  and 
consequently  it  will  be  necessary  merely  to  subtract 
the  given  divisor  from  r*  as  many  times  as  is  requisite 
to  obtain  a  positive  or  negative  remainder  less  than 


ON  THE  OPERATIONS  OF  ARITHMETIC.  37 

half  of  that  divisor.  Let  s  be  that  remainder ;  we  shall 
then  only  have  to  multiply  s  by  r,  the  remainder  from 
the  division  of  10,  to  obtain  the  remainder  from  the 
division  of  1000  by  the  given  divisor,  because  1000  is 
100  X  10,  and  so  on. 

For  example,  dividing  10  by  7  we  have  a  remainder 
of  3  ;  hence,  the  remainder  from  dividing  100  by  7 
will  be  9,  or,  subtracting  from  9  the  given  divisor  7, 
2.  The  remainder  from  dividing  1000  by  7,  then,  will 
be  the  product  of  2  by  3  or  6,  or,  subtracting  the  di- 
visor, 7,  — 1.  Again,  the  remainder  from  dividing 
10,000  by  7  will  be  the  product  of  —1  and  3,  or  —3, 
and  so  on. 

Let  us  now  take  the  divisor  11.     The  remainder 
from  dividing  1  by  11  is  1,  from  dividing  10  by  11  is  Test  of 
10,  or,  subtracting  the  divisor,  — 1.     The  remainder  by  „. 
from  dividing  100  by  11,  then,  will  be  the  square  of 
—  1,  or  1  ;  from  dividing  1000  by  11  it  will  be  1  mul- 
tiplied by  — 1  or — 1  again,  and  so  on  forever,  the  re- 
mainders forming  the  infinite  series 

1,  —1,  1,  —  1,  1,  —1,  ... 

Hence  results  the  remarkable  property  of  the  num- 
ber 11,  that  if  the  digits  of  any  number  be  alternately 
added  and  subtracted,  that  is  to  say,  if  we  take  the 
sum  of  the  first,  the  third,  and  the  fifth,  etc.,  and  sub- 
tract from  it  the  sum  of  the  second,  the  fourth,  the 
sixth,  etc.,  we  shall  obtain  the  remainder  which  re- 
sults from  dividing  that  number  by  the  number  11. 


38  ON  THE  OPERATIONS  OF  ARITHMETIC. 

The  preceding    theory  of   remainders    is    fraught 

with  remarkable  consequences,  and  has  given  rise  to 

Theory  of     many  ingenious  and  difficult  investigations.     We  can 

remainders  ,,.-,,... 

demonstrate,  tor  example,  that  it  the  divisor  is  a  prime 
number,  the  remainders  of  any  progression  1,  a,  a2, 
as,  a4,  .  .  .  form  periods  which  will  recur  continually 
to  infinity,  and  all  of  which,  like  the  first,  begin  with 
unity;  in  such  wise  that  when  unity  reappears  among 
the  remainders  we  may  continue  them  to  infinity  by 
simply  repeating  the  remainders  which  precede.  It 
has  also  been  demonstrated  that  these  periods  can 
only  contain  a  number  of  terms  which  is  equal  to  the 
divisor  less  1  or  to  an  aliquot  part  of  the  divisor  less 
1.  But  we  have  not  yet  been  able  to  determine  a  priori 
this  number  for  any  divisor  whatever. 

As  to  the  utility  of  this  method  for  finding  the  re- 
mainder resulting  from  dividing  a  given  number  by  a 
given  divisor,  it  is  frequently  very  useful  when  one 
has  several  numbers  to  divide  by  the  same  number, 
and  it  is  required  to  prepare  a  table  of  the  remainders. 
While  as  to  division  by  9  and  11,  since  that  is  very 
simple,  it  can  be  employed  as  a  check  upon  multipli- 
cation and  division.  Having  found  the  remainders 
from  dividing  the  multiplicand  and  the  multiplier  by 
either  of  these  numbers  it  is  simply  necessary  to  take 
the  product  of  the  two  remainders  so  resulting,  from 
which,  after  subtracting  the  divisor  as  many  times  as 
is  requisite,  we  shall  obtain  the  remainder  from  di- 
viding their  product  by  the  given  divisor, — a  remain- 


ON  THE  OPERATIONS  OF  ARITHMETIC.  39 

der  which  should  agree  with  the  remainder  obtained 
from  treating  the  actual  product  in  this  manner.  And 
since  in  division  the  dividend  less  the  remainder  should  Checks  on 

multiplica- 

be  equal  to  the  product  of  the  divisor  and  the  quo-  tion  and 

division. 

tient,  the  same  check  may  also  be  applied  here  to  ad- 
vantage. 

The  supposition  which  I  have  just  made  that  the 
product  of  the  remainders  from  dividing  two  numbers 
by  the  same  divisor  is  equal  to  the  remainder  from 
dividing  the  product  of  these  numbers  by  the  same 
divisor  is  easily  proved,  and  I  here  give  a  general 
demonstration  of  it. 

Let  M  and  N  be  two  numbers,  D  the  divisor,  p 
and  q  the  quotients,  and  r,  s  the  two  remainders.  We 
shall  plainly  have 

M=pD  +  r,  N=  qD  +  s, 
from  which  by  multiplying  we  obtain 

MN=pqD'1  +  spD+rqD-\-  rs: 

where  it  will  be  seen  that  all  the  terms  are  divisible 
by  D  with  the  exception  of  the  last,  rs,  whence  it  fol- 
lows that  r  s  will  be  the  remainder  from  dividing  MN 
by  D.  It  is  further  evident  that  if  any  multiple  what- 
ever of  D,  as  mD,  be  subtracted  from  rs,  the  result 
r s — mD  will  also  be  the  remainder  from  dividing  MN 
by  D.  For,  putting  the  value  of  MN  in  the  following 

form  : 

/  qD*  4-  spD  -\-rqD-\-  ?nD  -[-rs  —  mD, 

it  is  obvious  that  the  remaining  terms  are  all  divisible 


40  ON  THE  OPERATIONS  OF  ARITHMETIC. 

by  D.     And  this  remainder  r s  —  mD  can  always  be 

made  less  than  D,  or,  by  employing  negative  remain- 

D. 
ders,  less  even  than  --^- 

This  is  all  that  I  have  to  say  upon  multiplication 
Evolution,  and  division.  I  shall  not  speak  of  the  extraction  of 
roots.  The  rule  is  quite  simple  for  square  roots ;  it 
leads  directly  to  its  goal ;  trials  are  unnecessary.  As 
to  cube  and  higher  roots,  the  occasion  rarely  arises 
for  extracting  them,  and  when  it  does  arise  the  ex- 
traction can  be  performed  with  great  facility  by  means 
of  logarithms,  where  the  degree  of  exactitude  can  be 
carried  to  as  many  decimal  places  as  the  logarithms 
themselves  have  decimal  places.  Thus,  with  seven- 
place  logarithms  we  can  extract  roots  having  seven 
figures,  and  with  the  large  tables  where  the  loga- 
rithms have  been  calculated  to  ten  decimal  places  we 
can  obtain  even  ten  figures  of  the  result. 

One  of  the  most  important  operations  in  arith- 
metic is  the  so-called  rule  of  three,  which  consists  in 
finding  the  fourth  term  of  a  proportion  of  which  the 
first  three  terms  are  given. 

In  the  ordinary  text-books  of  arithmetic  this  rule 
has  been  unnecessarily  complicated,  having  been  di- 
vided into  simple,  direct,  inverse,  and  compound  rules 
of  three.  In  general  it  is  sufficient  to  comprehend  the 
conditions  of  the  problem  thoroughly,  for  the  common 
rule  of  three  is  always  applicable  where  a  quantity  in- 
creases or  diminishes  in  the  same  proportion  as  an- 


ON  THE  OPERATIONS  OF  ARITHMETIC.  41 

other.  For  example,  the  price  of  things  augments  in 
proportion  to  the  quantity  of  the  things,  so  that  the 
quantity  of  the  thing  being  doubled,  the  price  also  Rule  of 

three. 

will  be  doubled,  and  so  on.  Similarly,  the  amount  of 
work  done  increases  proportionally  to  the  number  of 
persons  employed.  Again,  things  may  increase  si- 
multaneously in  two  different  proportions.  For  ex- 
ample, the  quantity  of  work  done  increases  with  the 
number  of  the  persons  employed,  and  also  with  the 
time  during  which  they  are  employed.  Further,  there 
are  things  that  decrease  as  others  increase. 

Now  all  this  may  be  embraced  in  a  single,  simple 
proposition.  If  a  quantity  increases  both  in  the  ratio 
in  which  one  or  several  other  quantities  increase  and 
in  that  in  which  one  or  several  other  quantities  de- 
crease, it  is  the  same  thing  as  saying  that  the  proposed 
quantity  increases  proportionally  to  the  product  of  the 
quantities  which  increase  with  it,  divided  by  the  pro- 
duct of  the  quantities  which  simultaneously  decrease. 
For  example,  since  the  quantity  of  work  done  in- 
creases proportionally  with  the  number  of  laborers 
and  with  the  time  during  which  they  work  and  since 
it  diminishes  in  proportion  as  the  work  becomes  more 
difficult,  we  may  say  that  the  result  is  proportional  to 
the  number  of  laborers  multiplied  by  the  number 
measuring  the  time  during  which  they  labor,  divided 
by  the  number  which  measures  or  expresses  the  diffi- 
culty of  the  work. 

The  further  fact  should  not  be  lost  sight  of  that 


42  ON  THE  OPERATIONS  OF  ARITHMETIC. 

the  rule  of  three  is  properly  applicable  only  to  things 

which  increase  in  a  constant  ratio.    For  example,  it  is 

AppHcabii-  assumed  that  if  a  man  does  a  certain  amount  of  work 

ityofthe         .  ..,,.. 

rule  of  m  one  day,  two  men  will  do  twice  that  amount  in  one 
day,  three  men  three  times  that  amount,  four  men 
four  times  that  amount,  etc.  In  reality  this  is  not  the 
case,  but  in  the  rule  of  proportion  it  is  assumed  to  be 
such,  since  otherwise  we  should  not  be  able  to  em- 
ploy it. 

When  the  law  of  augmentation  or  diminution  va- 
ries, the  rule  of  three  is  not  applicable,  and  the  ordi- 
nary methods  of  arithmetic  are  found  wanting.  We 
must  then  have  recourse  to  algebra. 

A  cask  of  a  certain  capacity  empties  itself  in  a  cer- 
tain time.  If  we  were  to  conclude  from  this  that  a 
cask  of  double  that  capacity  would  empty  itself  in 
double  the  time,  we  should  be  mistaken,  for  it  will 
empty  itself  in  a  much  shorter  time.  The  law  of  ef- 
flux does  not  follow  a  constant  ratio  but  a  variable 
ratio  which  diminishes  with  the  quantity  of  liquid  re- 
maining in  the  cask. 

We  know  from  mechanics  that  the  spaces  traversed 
by  a  body  in  uniform  motion  bear  a  constant  ratio  to 
the  times  elapsed.  If  we  travel  one  mile  in  one  hour, 
in  two  hours  we  shall  travel  two  miles.  But  the  spaces 
traversed  by  a  falling  stone  are  not  in  a  fixed  ratio  to 
the  time.  If  it  falls  sixteen  feet  in  the  first  second,  it 
will  fall  forty-eight  feet  in  the  second  second. 

The  rule  of  three  is  applicable  when  the  ratios  are 


ON  THE  OPERATIONS  OF  ARITHMETIC.  43 

constant  only.  And  in  the  majority  of  affairs  of  ordin- 
ary life  constant  ratios  are  the  rule.  In  general,  the 
price  is  always  proportional  to  the  quantity,  so  that  if  Theory  and 
a  given  thing  has  a  certain  value,  two  such  things  will 
have  twice  that  value,  three  three  times  that  value, 
four  four  times  that  value,  etc.  It  is  the  same  with 
the  product  of  labor  relatively  to  the  number  of  labor- 
ers and  to  the  duration  of  the  labor.  Nevertheless, 
cases  occur  in  which  we  may  be  easily  led  into  error. 
If  two  horses,  for  example,  can  pull  a  load  of  a  cer- 
tain weight,  it  is  natural  to  suppose  that  four  horses 
could  pull  a  load  of  double  that  weight,  six  horses  a 
load  of  three  times  that  weight.  Yet,  strictly  speak- 
ing, such  is  not  the  case.  For  the  inference  is  based 
upon  the  assumption  that  the  four  horses  pull  alike  in 
amount  and  direction,  which  in  practice  can  scarcely 
ever  be  the  case.  It  so  happens  that  we  are  frequently 
led  in  our  reckonings  to  results  which  diverge  widely 
from  reality.  But  the  fault  is  not  the  fault  of  mathe- 
matics ;  for  mathematics  always  gives  back  to  us  ex- 
actly what  we  have  put  into  it.  The  ratio  was  constant 
according  to  the  supposition.  The  result  is  founded 
upon  that  supposition.  If  the  supposition  is  false  the 
result  is  necessarily  false.  Whenever  it  has  been  at- 
tempted to  charge  mathematics  with  inexactitude,  the 
accusers  have  simply  attributed  to  mathematics  the 
error  of  the  calculator.  False  or  inexact  data  having 
been  employed  by  him,  the  result  also  has  been  neces- 
sarily false  or  inexact. 


44  ON  THE  OPERATIONS  OF  ARITHMETIC. 

Among  the  other  rules  of  arithmetic  there  is  one 
called  alligation  which  deserves  special  consideration 
Alligation,  from  the  numerous  applications  which  it  has.  Al- 
though alligation  is  mainly  used  with  reference  to  the 
mingling  of  metals  by  fusion,  it  is  yet  applied  gener- 
ally to  mixtures  of  any  number  of  articles  of  different 
values  which  are  to  be  compounded  into  a  whole  of  a 
like  number  of  parts  having  a  mean  value.  The  rule 
of  alligation,  or  mixtures,  accordingly,  has  two  parts. 

In  one  we  seek  the  mean  and  common  value  of 
each  part  of  the  mixture,  having  given  the  number 
of  the  parts  and  the  particular  value  of  each.  In  the 
second,  having  given  the  total  number  of  the  parts 
and  their  mean  value,  we  seek  the  composition  of  the 
mixture  itself,  or  the  proportional  number  of  parts  of 
each  ingredient  which  must  be  mixed  or  alligated  to- 
gether. 

Let  us  suppose,  for  example,  that  we  have  several 
bushels  of  grain  of  different  prices,  and  that  we  are 
desirous  of  knowing  the  mean  price.  The  mean  price 
must  be  such  that  if  each  bushel  were  of  that  price  the 
total  price  of  all  the  bushels  together  would  still  be 
the  same.  Whence  it  is  easy  to  see  that  to  find  the 
mean  price  in  the  present  case  we  have  first  simply  to 
find  the  total  price  and  to  divide  it  by  the  number  of 
bushels. 

In  general  if  we  multiply  the  number  of  things  of 
each  kind  by  the  value  of  the  unit  of  that  kind  and 
then  divide  the  sum  of  all  these  products  by  the  total 


ON  THE  OPERATIONS  OF  ARITHMETIC.  45 

number  of  things,  we  shall  have  the  mean  value,  be- 
cause that  value  multiplied  by  the  number  of  the 
things  will  again  give  the  total  value  of  all  the  things 
taken  together. 

This  mean  or  average  value  as  it  is  called,  is  of 
great  utility  in  almost  all  the  affairs  of  life.  When-  Mean 

values. 

ever  we  arrive  at  a  number  of  different  results,  we 
always  like  to  reduce  them  to  a  mean  or  average  ex- 
pression which  will  yield  the  same  total  result. 

You  will  see  when  you  come  to  the  calculus  of 
probabilities  that  this  science  is  almost  entirely  based 
upon  the  principle  we  are  discussing. 

The  registration  of  births  and  deaths  has  rendered 
possible  the  construction  of  so-called  tables  of  mortality 
which  show  what  proportion  of  a  given  number  of 
children  born  at  the  same  time  or  in  the  same  year 
survive  at  the  end  of  one  year,  two  years,  three  years, 
etc.  So  that  we  may  ask  upon  this  basis  what  is  the 
mean  or  average  value  of  the  life  of  a  person  at  any 
given  age.  If  we  look  up  in  the  tables  the  number  of 
people  living  at  a  certain  age,  and  then  add  to  this 
the  number  of  persons  living  at  all  subsequent  ages, 
it  is  clear  that  this  sum  will  give  the  total  number  of 
years  which  all  living  persons  of  the  age  in  question 
have  still  to  live.  Consequently,  it  is  only  necessary 
to  divide  this  sum  by  the  number  of  living  persons  of 
a  certain  age  in  order  to  obtain  the  average  duration 
of  life  of  such  persons,  or  better,  the  number  of  years 
which  each  person  must  live  that  the  total  number  of 


46  ON  THE  OPERATIONS  OF  ARITHMETIC. 

years  lived  by  all  shall  be  the  same  and  that  each 
person  shall  have  lived  an  equal  number.  It  has  been 
Probability  found  in  this  manner  by  taking  the  mean  of  the  re- 
sults of  different  tables  of  mortality,  that  for  an  in- 
fant one  year  old  the  average  duration  of  life  is  about 
40  years ;  for  a  child  ten  years  old  it  is  still  40  years  ; 
for  20  it  is  34  ;  for  30  it  is  26  ;  for  40  it  is  23  ;  for  50 
it  is'  17  ;  for  GO  it  is  12  ;  for  70,  8  ;  and  for  80,  5. 

To  take  another  example,  a  number  of  different 
experiments  are  made.  Three  experiments  have  given 
4  as  a  result ;  two  experiments  have  given  5  ;  and  one 
has  given  6.  To  find  the  mean  we  multiply  4  by  3,  5 
by  2,  and  1  by  6,  add  the  products  which  gives  28, 
and  divide  28  by  the  number  of  experiments  or  6, 
which  gives  4|  as  the  mean  result  of  all  the  experi- 
ments. 

But  it  will  be  apparent  that  this  result  can  be  re- 
garded as  exact  only  upon  the  condition  of  our  having 
supposed  that  the  experiments  were  all  conducted  with 
equal  precision.  But  it  is  impossible  that  such  could 
have  been  the  case,  and  it  is  consequently  imperative 
to  take  account  of  these  inequalities,  a  requirement 
which  would  demand  a  far  more  complicated  calculus 
than  that  which  we  have  employed,  and  one  which  is 
now  engaging  the  attention  of  mathematicians. 

The  foregoing  is  the  substance  of  the  first  part  ot 
the  rule  of  alligation  ;  the  second  part  is  the  opposite 
of  the  first.  Given  the  mean  value,  to  find  how  much 


ON  THE  OPERATIONS  OF  ARITHMETIC.  47 

must  be  taken  of  each  ingredient  to  produce  the  re- 
quired mean  value. 

The  problems  of  the  first  class  are  always  deter- 
minate, because,  as  we  have  just  seen,  the  number  of  Alternate 
units  of  each  ingredient  has  simply  to  be  multiplied 
by  the  value  of  each  ingredient  and  the  sum  of  all 
these  products  divided  by  the  number  of  the  ingredi- 
ents. 

The  problems  of  the  second  class,  on  the  other 
hand,  are  always  indeterminate.  But  the  condition 
that  only  positive  whole  numbers  shall  be  admitted 
in  the  result  serves  to  limit  the  number  of  the  solu- 
tions. 

Suppose  that  we  have  two  kinds  of  things,  that 
the  value  of  the  unit  of  one  kind  is  a,  and  that  of  the 
unit  of  the  second  is  b,  and  that  it  is  required  to  find 
how  many  units  of  the  first  kind  and  how  many  units 
of  the  second  must  be  taken  to  form  a  mixture  or 
whole  of  which  the  mean  value  shall  be  m. 

Call  x  the  number  of  units  of  the  first  kind  that 
must  enter  into  the  mixture,  and  y  the  number  of  units 
of  the  second  kind.  It  is  clear  that  ax  will  be  the 
value  of  the  x  units  of  the  first  kind,  and  by  the  value 
of  the  y  units  of  the  second.  Hence  ax-\-  by  will  be 
the  total  value  of  the  mixture.  But  the  mean  value 
of  the  mixture  being  by  supposition  m,  the  sum  x-\-y 
of  the  units  of  the  mixture  multiplied  by  m,  the  mean 
value  of  each  unif,  must  give  the  same  total  value. 
We  shall  have,  therefore,  the  equation 


48  ON  THE  OPERATIONS  OF  ARITHMETIC. 

a  x  -\-  by  =  m  x  -\-  my. 

Transposing  to  one  side  the  terms  multiplied  by  x 
and  to  the  other  the  terms  multiplied  by  y,  we  obtain  : 

Two  in-  f  TN 

gradients.  (0  — *)*=  (»—*).* 

and  dividing  by  a  —  m  we  get 


whence  it  appears  that  the  number  y  may  be  taken  at 
pleasure,  for  whatever  be  the  value  given  to  y,  there 
will  always  be  a  corresponding  value  of  x  which  will 
satisfy  the  problem. 

Such  is  the  general  solution  which  algebra  gives. 
But  if  the  condition  be  added  that  the  two  numbers  x 
and  y  shall  be  integers,  then  y  may  not  be  taken  at 
pleasure.  In  order  to  see  how  we  can  satisfy  this  last 
condition  in  the  simplest  manner,  let  us  divide  the 
last  equation  by_y,  and  we  shall  have 

x       m  —  b 
y       a  —  m 

For  x  and  y  both  to  be  positive,  it  is  necessary  that 

the  quantities 

m  —  b  and  a  —  m 

should  both  have  the  same  sign ;  that  is  to  say,  if  a  is 
greater  or  less  than  m,  then  conversely  b  must  be  less 
or  greater  than  m;  or  again,  m  must  lie  between  a 
and  b,  which  is  evident  from  the  condition  of  the 
problem.  Suppose  a,  then,  to  be  the  greater  and  b 


mixtures. 


ON  THE  OPERATIONS  OF  ARITHMETIC.  49 

the  smaller  of  the  two  prices.  It  remains  to  find  the 
value  of  the  fraction 

m b  Rule  of 

a  —  m' 
which  if  necessary  is  to  be  reduced  to  its  lowest  terms. 

73 

Let  •  -  -  be  that  fraction  reduced  to  its  lowest  terms.  It 
is  clear  that  the  simplest  solution  will  be  that  in  which 

x  =  B  and  y  =  A, 

But  since  a  fraction  is  not  altered  by  multiplying  its 
numerator  and  denominator  by  the  same  number,  it 
is  clear  that  we  may  also  take  x  =  nB  andy  =  nA,  n 
being  any  number  whatever,  provided  it  is  an  integer, 
for  by  supposition  x  and  y  must  be  integers.  And  it 
is  easy  to  prove  that  these  expressions  of  x  and  y  are 
the  only  ones  which  will  resolve  the  proposed  prob- 
lem. According  to  the  ordinary  rule  of  mixtures,  x, 
the  quantity  of  the  dearer  ingredient,  is  made  equal 
to  m  —  b,  the  excess  of  the  average  price  above  the 
lower  price,  and  y  the  quantity  of  the  cheaper  ingre- 
dient is  made  equal  to  a  —  m,  the  excess  of  the  higher 
price  above  the  average  price,  —  a  rule  which  is  con- 
tained directly  in  the  general  solution  above  given. 

Suppose,  now,  that  instead  of  two  kinds  of  things, 
we  have  three  kinds,  the  values  of  which  beginning 
with  the  highest  are  a,  b,  and  c.  Let  x,  y,  z  be  the 
quantities  which  must  be  taken  of  each  to  form  a  mix- 
ture or  compound  having  the  mean  value  m.  The 
sum  of  the  values  of  the  three  quantities  x,  y,  z  will 
then  be 


50  ON  THE  OPERATIONS  OF  ARITHMETIC. 

But  this  total  value  must  be  the  same  as  that  pro- 
duced if  all  the  individual  values  were  m,  in  which 
Three  in-     case  the  total  value  is  obviously 

gredients. 

m  x  -j-  my  -f-  m  z. 
The  following  equation,  therefore,  must  be  satisfied : 

ax-\-by-\-cz^=mx-\-  my -\-rnz, 
or,  more  simply, 

(a  —  m)x-\-  (b  —  m)y  -f-  (c  —  m)  z  =  0. 
Since  there  are  three  unknown  quantities  in  this  equa- 
tion, two  of  them  may  be  taken  at  pleasure.  But  if 
the  condition  is  that  they  shall  be  expressed  by  posi- 
tive integers,  it  is  to  be  observed  first  that  the  num- 
bers 

a  —  m  and  ;// —  c 

are  necessarily  positive ;  so  that  putting  the  equation 
in  the  form 

(a  —  m)  x  —  (m  —  c)z=(nt  —  b}y, 

the  question  resolves  itself  into  finding  two  multiples 
of  the  given  numbers 

a  —  m  and  m  —  c 
whose  difference  shall  be  equal  to  (;//  —  b}y. 

This  question  is  always  resolvable  in  whole  num- 
bers whatever  the  given  numbers  be  of  which  we  seek 
the  multiples,  and  whatever  be  the  difference  between 
these  multiples.  As  it  is  sufficiently  remarkable  in  it- 
self and  may  be  of  utility  in  many  emergencies,  we 
shall  give  here  a  general  solution  of  it,  derived  from 
the  properties  of  continued  fractions. 


ON  THE  OPERATIONS  OF  ARITHMETIC.  5! 

Let  M  and  N  be  two  whole  numbers.  Of  these 
numbers  two  multiples  xM,  zN  are  sought  whose  dif- 
ference is  given  and  equal  to  D.  The  following  equa-  General 

•11  i  •    e-  solution. 

tion  will  then  have  to  be  satisfied 


where  x  and  z  by  supposition  are  whole  numbers.  In 
the  first  place,  it  is  plain  that  if  M  and  N  are  not 
prime  to  each  other,  the  number  D  is  divisible  by  the 
greatest  common  divisor  of  M  and  N;  and  the  divis- 
ion having  been  performed,  we  should  have  a  similar 
equation  in  which  the  numbers  M  and  N  are  prime 
to  each  other,  so  that  we  are  at  liberty  always  to  sup- 
pose them  reduced  to  that  condition.  I  now  observe 
that  if  we  know  the  solution  of  the  equation  for  the 
case  in  which  the  number  D  is  equal  to  -j-  1  or  —  1, 
we  can  deduce  the  solution  of  it  for  any  value  what- 
ever of  D.  For  example,  suppose  that  we  know  two 
multiples  of  AT  and  N,  say  pM  and  qN,  the  difference 
of  which  pM  —  qN  is  equal  to  ±1.  Then  obviously 
we  shall  merely  have  to  multiply  both  these  multiples 
by  the  number  D  to  obtain  a  difference  equal  to  ±D. 
For,  multiplying  the  preceding  equation  by  D,  we 

have 

pDM—  qDN=  ±  D  ; 

and  subtracting  the  latter  equation  from  the  original 

equation 

xM—  zN=  D, 

or  adding  it,  according  as  the  term  D  has  the  sign 
-|-  or  —  before  it,  we  obtain 


52  ON  THE  OPERATIONS  OF  ARITHMETIC. 

(x  =p/Z>)  M—  (z  zp  qD}  N=  0, 

which  gives  at  once,  as  we  saw  above  in  the  rule  for 
the  mixture  of  two  different  ingredients, 

x  =p  pD  =  nN,  z  q=  qD  =  nM, 

n  being  any  number  whatever.  So  that  we  have  gen- 
erally 

x  =  nJV±  pD  and  z  =  nM±  qD 

where  n  is  any  whole  number,  positive  or  negative. 
It  remains  merely  to  find  two  numbers  /  and  q  such 
that 


Now  this  question  is  easily  resolvable  by  continued 
fractions.  For  we  have  seen  in  treating  of  these  frac- 
tions that  if  the  fraction  —  be  reduced  to  a  continued 
fraction,  and  all  the  successive  fractions  approximat- 

ing to  its  value  be  calculated,  the  last  of  these  succes- 

M 
sive  fractions  being  the  fraction  —  itself,  then  the  se- 

ries of  fractions  so  reached  is  such  that  the  difference 
between  any  two  consecutive  fractions  is  always  equal 
to  a  fraction  of  which  the  numerator  is  unity  and  the 

denominator  the   product  of  the   two  denominators. 

tr 
For  example,    designating   by        the    fraction    which 

immediately  precedes  the  last  fraction  —  we   obtain 

necessarily 

LM—KN=\  or  —  1, 

M  .  K  . 

according   as  -^=  is    greater  or  less   than      ,  in  other 


ON  THE  OPERATIONS  OF  ARITHMETIC.  53 

words,  according  as  the  place  occupied  by  the  last 

M 

fraction  —  in  the  series  of  fractions  successively  ap- 
proximating to  its  value  is  even  or  odd ;  for,  the  first  Resolution 
fraction  of  the  approximating  series  is  always  smaller,  Ued°f"a"n 
the  second  larger,   the  third  smaller,  etc.,  than  the  tlons- 
original  fraction  which  is  identical  with  the  last  frac- 
tion of  the  series.     Making,  therefore, 

p  =  L  and  q  =  K, 

the  problem  of  the  two  multiples  will  be  resolved  in 
all  its  generality. 

It  is  now  clear  that  in  order  to  apply  the  foregoing 
solution  to  the  initial  question  regarding  alligation  we 
have  simply  to  put 

J/—  a  —  m,  7V=  m  —  c,  and  D  =  (m  —  b~)y  ; 
so  that  the  number  y  remains  undetermined  and  may 
be  taken  at  pleasure,  as  may  also  the  number  TV  which 
appears  in  the  expressions  for  x  and  z. 


LECTURE  III. 


ON  ALGEBRA,   PARTICULARLY  THE    RESOLUTION  OF 

EQUATIONS  OF  THE  THIRD  AND 

FOURTH  DEGREE. 


A  LGEBRA  is  a  science  almost  entirely  due  to  the 
-^~~*-     moderns.      I  say  almost  entirely,  for  we  have 
Algebra       one  treatise  from  the  Greeks,  that  of  Diophantus,  who 
Indents  °    flourished  in  the  third*  century  of  the  Christian  era. 
This  work  is  the  only  one  which  we  owe  to  the  an- 
cients in  this  branch  of  mathematics.     When  I  speak 
of  the  ancients  I  speak  of  the  Greeks  only,  for  the 
Romans  have  left  nothing  in  the  sciences,  and  to  all 
appearances  did  nothing. 

Diophantus  may  be  regarded  as  the  inventor  of 
algebra. f  From  a  word  in  his  preface,  or  rather  in  his 
letter  of  dedication,  (for  the  ancient  geometers  were 
wont  to  address  their  productions  to  certain  of  their 
friends,  a  practice  exemplified  in  the  prefaces  of  Apol- 
lonius  and  Archimedes),  from  a  word  in  his  preface,  I 
say,  we  learn  that  he  was  the  first  to  occupy  himself 

*The  period  is  uncertain.     Some  say  in  the  fourth  century.     See  Cantor, 
Gcschichte  dcr  .Matheinatik,  2nd.  ed.,  Vol.  I.,  p.  434. —  Trans, 
t  On  this  point,  see  Appendix,  p.  151. —  Trans, 


ON  ALGEBRA.  55 

with  that  branch  of  arithmetic  which  has  since  been 
called  algebra. 

His  work  contains  the  first  elements  of  this  science. 
He  employed  to  express  the  unknown  quantity  a  Greek  Diophantus 
letter  which  corresponds  to  our  j/*  and  which  has 
been  replaced  in  the  translations  by  N.  To  express 
the  known  quantities  he  employed  numbers  solely,  for 
algebra  was  long  destined  to  be  restricted  entirely  to 
the  solution  of  numerical  problems.  We  find,  how- 
ever, that  in  setting  up  his  equations  consonantly  with 
the  conditions  of  the  problem  he  uses  the  known  and 
the  unknown  quantities  alike.  And  herein  consists 
virtually  the  essence  of  algebra,  which  is  to  employ 
unknown  quantities,  to  calculate  with  them  as  we  do 
with  known  quantities,  and  to  form  from  them  one 
or  several  equations  from  which  the  value  of  the  un- 
known quantities  can  be  determined.  Although  the 
work  of  Diophantus  contains  indeterminate  problems 
almost  exclusively,  the  solution  of  which  he  seeks  in 
rational  numbers, — problems  which  have  been  desig- 
nated after  him  Diophantine problems, — we  nevertheless 
find  in  his  work  the  solution  of  a  number  of  determi- 
nate problems  of  the  first  degree,  and  even  of  such 
as  involve  several  unknown  quantities.  In  the  latter 
case,  however,  the  author  invariably  has  recourse  to 
particular  artifices  for  reducing  the  problem  to  a  single 
unknown  quantity, — which  is  not  difficult.  He  gives, 

*According  to  a  recent  conjecture,  the  character  in  question  is  an  abbre- 
viation of  ap  the  first  letters  of  apefyxos,  number,  the  appellation  technically 
applied  by  Diophantus  to  the  unknown  quantity.—  Trans. 


56  ON  ALGEBRA. 

also,  the  solution  of  equations  of  the  second  degree,  but 
is  careful  so  to  arrange  them  that  they  never  assume 
the  affected  form  containing  the  square  and  the  first 
power  of  the  unknown  quantity. 

He  proposed,  for  example,  the  following  question 
Equations    which  involves  the  general  theory  of  equations  of  the 

of  the  sec-  ,     , 

ond  degree,  second  degree  : 

To  find  two  numbers  the  sum  and  the  product  of  which 
are  given. 

If  we  call  the  sum  a  and  the  product  b  we  have  at 
once,  by  the  theory  of  equations,  the  equation 

iX2_0iX,_j_0_0. 

Diophantus  resolves  this  problem  in  the  following 
manner.  The  sum  of  the  two  numbers  being  given, 
he  seeks  their  difference,  and  takes  the  latter  as  the 
unknown  quantity.  He  then  expresses  the  two  num- 
bers  in  terms  of  their  sum  and  difference, — the  one 
by  half  the  sum  plus  half  the  difference,  the  other  by 
half  the  sum  less  half  the  difference, —  and  he  has 
then  simply  to  satisfy  the  other  condition  by  equating 
their  product  to  the  given  number.  Calling  the  given 
sum  a,  the  unknown  difference  x,  one  of  the  numbers 

/J          I          *£  ft  J£ 

will  be  -      —  and  the  other  will  be  -      —  •     Multiply- 

ai x-2 

ing  these  together  we  have  — The  term  con- 
taining x  is  here  eliminated,  and  equating  the  quan- 
tity last  obtained  to  the  given  product,  we  have  the 

simple  equation 

a2  —  x* 

~~A —"> 


ON  ALGEBRA.  57 

from  which  we  obtain 

xi  =  a?  —  46, 
and  from  the  latter 


Diophantus  resolves  several  other  problems  of  this 
class.  By  appropriately  treating  the  sum  or  differ-  other  prob- 

.  .  •       lems  solved 

ence  as  the  unknown  quantity  he  always  arrives  at  an  by  Dio. 
equation  in  which  he  has  only  to  extract  a  square  root  phantus- 
to  reach  the  solution  of  his  problem. 

But  in  the  books  which  have  come  down  to  us 
(for  the  entire  work  of  Diophantus  has  not  been  pre- 
served) this  author  does  not  proceed  beyond  equa- 
tions of  the  second  degree,  and  we  do  not  know  if  he 
or  any  of  his  successors  (for  no  other  work  on  this 
subject  has  been  handed  down  from  antiquity)  ever 
pushed  their  researches  beyond  this  point. 

I  have  still  to  remark  in  connexion  with  the  work 
of  Diophantus  that  he  enunciated  the  principle  that 
-f-  and  —  give  —  in  multiplication,  and  —  and  —  ,  -)-, 
in  the  form  of  a  definition.  But  I  am  of  opinion  that 
this  is  an  error  of  the  copyists,  since  he  is  more  likely 
to  have  considered  it  as  an  axiom,  as  did  Euclid  some 
of  the  principles  of  geometry.  However  that  may  be, 
it  will  be  seen  that  Diophantus  regarded  the  rule  of 
the  signs  as  a  self-evident  principle  not  in  need  of  de- 
monstration. 

The  work  of  Diophantus  is  of  incalculable  value 
from  its  containing  the  first  germs  of  a  science  which 
because  of  the  enormous  progress  which  it  has  since 


58  ON  ALGEBRA. 

made  constitutes  one  of  the  chiefest  glories  of  the  hu- 
man intellect.     Diophantus  was  not  known  in  Europe 
Trans-         until  the  end  of  the  sixteenth  century,  the  first  transla- 
having  been  a  wretched  one  by  Xylander  made 


in  1575  and  based  upon  a  manuscript  found  about  the 
middle  of  the  sixteenth  century  in  the  Vatican  library, 
where  it  had  probably  been  carried  from  Greece  when 
the  Turks  took  possession  of  Constantinople. 

Bachet  de  Me'ziriac,  one  of  the  earliest  members 
of  the  French  Academy,  and  a  tolerably  good  mathe- 
matician for  his  time,  subsequently  published  (1621) 
a  new  translation  of  the  work  of  Diophantus  accom- 
panied by  lengthy  commentaries,  now  superfluous. 
Bachet's  translation  was  afterwards  reprinted  with  ob- 
servations and  notes  by  Fermat,  one  of  the  most  cel- 
ebrated mathematicians  of  France,  who  flourished 
about  the  middle  of  the  seventeenth  century,  and  of 
whom  we  shall  have  occasion  to  speak  in  the  sequel 
for  the  important  discoveries  which  he  has  made  in 
analysis.  Fermat's  edition  bears  the  date  of  1670.* 

It  is  much  to  be  desired  that  good  translations 
should  be  made,  not  only  of  the  work  of  Diophantus, 
but  also  of  the  small  number  of  other  mathematical 
works  which  the  Greeks  have  left  us.f 

*There  have  since  been  published  a  new  critical  edition  of  the  text  by 
M.  Paul  Tannery  (Leipsic,  1893  ,  and  two  German  translations,  one  by  O. 
Schulz  'Berlin,  1822)  and  one  by  G.  Wertheim  (Leipsic,  1890;.  Fermat's  notes 
on  Diophantus  have  been  republished  in  Vol.  I.  of  the  new  edition  of  Fermat's 
works  (Paris,  Gauthier-Villars  et  Fils,  1891).  —  Trans. 

t  Since  Lagrange's  time  this  want  has  been  partly  supplied.  Not  to  men- 
tion Euclid,  we  have,  for  example,  of  Archimedes  the  German  translation  of 
Nizze  (Stralsund,  1824)  and  the  French  translation  of  Peyrard  (Paris,  1807)  ;  oi 


ON  ALGEBRA.  59 

Prior  to  the  discovery  and  publication  of  Diophan- 
tus,  however,  algebra  had  already  found  its  way  into 
Europe.  Towards  the  end  of  the  fifteenth  century 
there  appeared  in  Venice  a  work  by  an  Italian  Fran- 
ciscan monk  named  Lucas  Paciolus  on  arithmetic  and 
geometry  in  which  the  elementary  rules  of  algebra 
were  stated.  This  book  was  published  (1494)  in  the  Algebra 
early  days  of  the  invention  of  printing,  and  the  fact  A'"a°bns8 
that  the  name  of  algebra  was  given  to  the  new  science 
shows  clearly  that  it  came  from  the  Arabs.  It  is  true 
that  the  signification  of  this  Arabic  word  is  still  dis- 
puted, but  we  shall  not  stop  to  discuss  such  matters, 
for  they  are  foreign  to  our  purpose.  Let  it  suffice 
that  the  word  has  become  the  name  for  a  science  that 
is  universally  known,  and  that  there  is  not  the  slight- 
est ambiguity  concerning  its  meaning,  since  up  to  the 
present  time  it  has  never  been  employed  to  designate 
anything  else. 

We  do  not  know  whether  the  Arabs  invented  alge- 
bra themselves  or  whether  they  took  it  frpm  the 
Greeks.*  There  is  reason  to  believe  that  they  pos- 
sessed the  work  of  Diophantus,  for  when  the  ages  of 
barbarism  and  ignorance  which  followed  their  first 
conquests  had  passed  by,  they  began  to  devote  them- 
selves to  the  sciences  and  to  translate  into  Arabic  all 
the  Greek  works  which  treated  of  scientific  subjects. 
It  is  reasonable  to  suppose,  therefore,  that  they  also 

Apollonius,  several  translations;  also  modern  translations  of  Hero,  Ptolemy, 
Pappus,  Theon,  Proclus,  and  several  others. 
*  See  Appendix,  p.  152. 


60  ON   ALGEBRA. 

translated  the  work  of  Diophantus  and  that  the  same 
work  stimulated  them  to  push  their  inquiries  farther 
in  this  science. 

Be  that  as  it  may,  the  Europeans,  having  received 
Algebra  in    algebra  from  the  Arabs,  were  in  possession  of  it  one 

Europe.      . 

hundred  years  before  the  work  of  Diophantus  was 
known  to  them.  They  made,  however,  no  progress 
beyond  equations  of  the  first  and  second  degree.  In 
the  work  of  Paciolus,  which  we  mentioned  above,  the 
general  resolution  of  equations  of  the  second  degree, 
such  as  we  now  have  it,  was  not  given.  We  find  in 
this  work  simply  rules,  expressed  in  bad  Latin  verses, 
for  resolving  each  particular  case  according  to  the 
different  combinations  of  the  signs  of  the  terms  of 
equation,  and  even  these  rules  applied  only  to  the 
case  where  the  roots  were  real  and  positive.  Negative 
roots  were  still  regarded  as  meaningless  and  superflu- 
ous. It  was  geometry  really  that  suggested  to  us  the 
use  of  negative  quantities,  and  herein  consists  one  of 
the  greatest  advantages  that  have  resulted  from  the 
application  of  algebra  to  geometry, — a  step  which  we 
owe  to  Descartes. 

In  the  subsequent  period  the  resolution  of  equations 
of  the  third  degree  was  investigated  and  the  discovery 
for  a  particular  case  ultimately  made  by  a  mathemati- 
cian of  Bologna  named  Scipio  Ferreus  (1515)-*  Two 
other  Italian  mathematicians,  Tartaglia  and  Cardan, 

*The  date  is  uncertain.     Tartaglia  gives  1506,  Cardan  1515.     Cantor  pre- 
fers the  latter.—  Trans. 


ON  ALGEBRA.  6l 

subsequently  perfected   the   solution   of  Ferreus  and 
rendered  it  general  for  all  equations  of  the  third  de- 
gree.     At  this  period,  Italy,  which  was  the  cradle  of  Tartagiia 
algebra  in  Europe,  was  still  almost  the  sole  cultivator  c^danW) 
of  the  science,  and  it  was  not   until   about  the  middle  to01-^}. 
of  the  sixteenth  century  that  treatises  on  algebra  be- 
gan  to  appear  in  France,  Germany,  and  other  coun- 
tries.    The  works  of  Peletier  and  Buteo  were  the  first 
which  France  produced  in  this  science,  the  treatise  of 
the  former  having  been  printed  in  1554  and  that  of 
the  latter  in  1559. 

Tartagiia  expounded  his  solution  in  bad  Italian 
verses  in  a  work  treating  of  divers  questions  and  in- 
ventions printed  in  1546,  a  work  which  enjoys  the 
distinction  of  being  one  of  the  first  to  treat  of  modern 
fortifications  by  bastions. 

About  the  same  time  (1545)  Cardan  published  his 
treatise  Ars  Magna,  or  Algebra,  in  which  he  left 
scarcely  anything  to  be  desired  in  the  resolution  of 
equations  of  the  third  degree.  Cardan  was  the  first  to 
perceive  that  equations  had  several  roots  and  to  dis- 
tinguish them  into  positive  and  negative.  But  he  is 
particularly  known  for  having  first  remarked  the  so- 
called  irreducible  case  in  which  the  expression  of  the 
real  roots  appears  in  an  imaginary  form.  Cardan  con- 
vinced himself  from  several  special  cases  in  which  the 
equation  had  rational  divisors  that  the  imaginary  form 
did  not  prevent  the  roots  from  having  a  real  value. 
But  it  remained  to  be  proved  that  not  only  were  the 


62  ON  ALGEBRA. 

roots  real  in  the  irreducible  case,  but  that  it  was  im- 
possible for  all  three  together  to  be  real  except  in  that 
case.  This  proof  was  afterwards  supplied  by  Vieta, 
and  particularly  by  Albert  Girard,  from  considerations 
touching  the  trisection  of  an  angle. 

We  shall  revert  later  on  to  the  irreducible  case  of 
The  equations  of  the  third  degree,  not  solely  because  it  pre- 

irreducible  r  r       i        i_       •       i  •  i   •    i. 

case  sents  a  new  form   of  algebraical  expressions  which 

have  found  extensive  application  in  analysis,  but  be- 
cause it  is  constantly  giving  rise  to  unprofitable  in- 
quiries with  a  view  to  reducing  the  imaginary  form  to 
a  real  form  and  because  it  thus  presents  in  algebra  a 
problem  which  may  be  placed  upon  the  same  footing 
with  the  famous  problems  of  the  duplication  of  the 
cube  and  the  squaring  of  the  circle  in  geometry. 

The  mathematicians  of  the  period  under  discus- 
sion were  wont  to  propound  to  one  another  problems 
for  solution.  These  problems  were  in  the  nature  of 
public  challenges  and  served  to  excite  and  to  main- 
tain in  the  minds  of  thinkers  that  fermentation  which 
is  necessary  for  the  pursuit  of  science.  The  challenges 
in  question  were  continued  down  to  the  beginning  of 
the  eighteenth  century  by  the  foremost  mathemati- 
cians of  Europe,  and  really  did  not  cease  until  the  rise 
of  the  Academies  which  fulfilled  the  same  end  in  a 
manner  even  more  conducive  to  the  progress  of  sci- 
ence, partly  by  the  union  of  the  knowledge  of  their 
various  members,  partly  by  the  intercourse  which  they 
maintained  between  them,  and  not  least  by  the  publi- 


ON  ALGEBRA.  63 

cation  of  their  memoirs,  which  served  to  disseminate 
the  new  discoveries  and  observations  among  all  per- 
sons interested  in  science. 

The  challenges  of  which  we  speak  supplied  in  a 
measure  the  lack  of  Academies,  which  were  not  yet  Biquadratic 

equations. 

in  existence,  and  we  owe  to  these  passages  at  arms 
many  important  discoveries  in  analysis.  Such  was 
the  resolution  of  equations  of  the  fourth  degree,  which 
was  propounded  in  the  following  problem. 

To  find  three  numbers  in  continued  proportion  of  which 
the  sum  is  10,  and  the  product  of  the  first  two  6. 

Generalising  and  calling  the  sum  of  the  three  num- 
bers a,  the  product  of  the  first  two  b,  and  the  first  two 
numbers  themselves  x,  y,  we  shall  have,  first,  xy^=b. 
Owing  to  the  continued  proportion,  the  third  number 

y2 

will  then  be  expressed  by—,  so  that  the  remaining 
condition  will  give 

y2 

x  -(-  y  •-)-  -----  =a. 

oc 

From  the  first  equation  we  obtain  x=—  .   which   sub- 

y 

stituted  in  the  second  gives 


Removing  the  fractions  and  arranging  the  terms,  we 

get  finally 

y  _j_  b  yi  ._  a  by  +  ifl  _  o, 

an  equation  of  the  fourth  degree  with  the  second  term 
missing. 

According  to  Bombelli,  of  whom  we  shall  speak 


64  ON  ALGEBRA. 

again,  Louis  Ferrari  of  Bologna  resolved  the  prob- 
lem by  a  highly  ingenious  method,  which  consists  in 
Ferrari  dividing  the  equation  into  two  parts  both  of  which 
BombeiH.  permit  of  the  extraction  of  the  square  root.  To  do 
this  it  is  necessary  to  add  to  the  two  numbers  quan- 
tities whose  determination  depends  on  an  equation  of 
the  third  degree,  so  that  the  resolution  of  equations 
of  the  fourth  degree  depends  upon  the  resolution  of 
equations  of  the  third  and  is  therefore  subject  to  the 
same  drawbacks  of  the  irreducible  case. 

The  Algebra  of  Bombelli  was  printed  in  Bologna 
in  1579*  in  the  Italian  language.  It  contains  not  only 
the  discovery  of  Ferrari  but  also  divers  other  impor- 
tant remarks  on  equations  of  the  second  and  third 
degree  and  particularly  on  the  theory  of  radicals  by 
means  of  which  the  author  succeeded  in  several  cases 
in  extracting  the  imaginary  cube  roots  of  the  two 
binomials  of  the  formula  of  the  third  degree  in  the  ir- 
reducible case,  so  finding  a  perfectly  real  result  and 
furnishing  thus  the  most  direct  proof  possible  of  the 
reality  of  this  species  of  expressions. 

Such  is  a  succinct  history  of  the  first  progress  of 
algebra  in  Italy.  The  solution  of  equations  of  the 
third  and  fourth  degree  was  quickly  accomplished. 
But  the  successive  efforts  of  mathematicians  for  over 
two  centuries  have  not  succeeded  in  surmounting  the 
difficulties  of  the  equation  of  the  fifth  degree. 

*  This  was  the  second  edition.  The  first  edition  appeared  in  Venice  in 
1572. —  Trans. 


ON  ALGEBRA.  65 

Yet  these  efforts  are  far  from  having  been  in  vain. 
They  have  given  rise  to  the  many  beautiful  theorems 
which  we  possess  on  the  formation  of  equations,  on  Theory  of 

equations. 

the  character  and  signs  of  the  roots,  on  the  trans- 
formation of  a  given  equation  into  others  of  which  the 
roots  may  be  formed  at  pleasure  from  the  roots  of  the 
given  equation,  and  finally,  to  the  beautiful  consider- 
ations concerning  the  metaphysics  of  the  resolution 
of  equations  from  which  the  most  direct  method  of 
arriving  at  their  solution,  when  possible,  has  resulted. 
All  this  has  been  presented  to  you  in  previous  lec- 
tures and  would  leave  nothing  to  be  desired  if  it  were 
but  applicable  to  the  resolution  of  equations  of  higher 
degree. 

Vieta  and  Descartes  in  France,  Harriot  in  Eng- 
land, and  Hudde  in  Holland,  were  the  first  after  the 
Italians  whom  we  have  just  mentioned  to  perfect  the 
theory  of  equations,  and  since  their  time  there  is 
scarcely  a  mathematician  of  note  that  has  not  applied 
himself  to  its  investigation,  so  that  in  its  present  state 
this  theory  is  the  result  of  so  many  different  inquiries 
that  it  is  difficult  in  the  extreme  to  assign  the  author 
of  each  of  the  numerous  discoveries  which  consti- 
tute it. 

I  promised  to  revert  to  the  irreducible  case.  To 
this  end  it  will  be  necessary  to  recall  the  method 
which  seems  to  have  led  to  the  original  resolution  of 
equations  of  the  third  degree  and  which  is  still  em- 
ployed in  the  majority  of  the  treatises  on  algebra. 


66  ON  ALGEBRA. 

Let  us  consider  the  general  equation  of  the  third  de- 
gree deprived  of  its  second  term,  which  can  always  be 
removed  ;  in  a  word,  let  us  consider  the  equation 


Equations 

of  the  third  Suppose 


=  0. 


where  y  and  z  are  two  new  unknown  quantities,  of 
which  one  consequently  may  be  taken  at  pleasure  and 
determined  as  we  think  most  convenient.  Substitut- 
ing this  value  for  x,  we  obtain  the  transformed  equation 


Factoring  the  two  terms  3y^z-\-  3_yz2  we  get 


and  the  transformed  equation  may  be  written  as  fol- 

lows : 

23       3,  z 


Putting  the  factor  multiplying  jy-f-  2  equal  to  zero,  — 
which  is  permissible  owing  to  the  two  undetermined 
quantities  involved,  —  we  shall  have  the  two  equations 


and 

ys  -f-  z3  -j-  q  =-.  0. 

from  which  y  and  z  can  be  determined.  The  means 
which  most  naturally  suggests  itself  to  this  end  is  to 
take  from  the  first  equation  the  value  of  z, 

z-    -P 
3/ 

and  to  substitute  it  in  the  second  equation,  removing 
the  fractions  by  multiplication.  So  proceeding,  we 


ON  ALGEBRA.  67 

obtain  the  following  equation  of  the  sixth  degree  in 
y,  called  the  reduced  equation, 

/  +  ?/-£  =  0,  The 

—  <  reduced 

which,  since  it  contains  two  powers  only  of  the  un-  e 
known  quantity,  of  which   one   is   the  square  of  the 
other,  is  resolvable  after  the  manner  of  equations  of 
the  second  degree  and  gives  immediately 


V6  = 4-  A  --  4-  • 

2       \  4    '   27 

from  which,  by  extracting  the  cube  root,  we  get 
and  finally, 


- 

This  expression  for  x  may  be  simplified  by  remarking 
that  the  product  of  y  by  the  radical 


»L_J 

^       2 


supposing  all  the  quantities  under  the  sign  to  be  mul- 
tiplied together,  is 


_3| 

27  3 


fi 

The  term^-,  accordingly,  takes  the  form 
6y 


«        q           \f_+f 

\       2  ~    \4  +27' 


and  we  have 


F        3         9          If     ,    P 
27 +  \  —  -»—  \T  +  27' 


68  ON  ALGEBRA. 

an  expression  in  which  the  square  root  underneath  the 
cubic  radical  occurs  in  both  its  plus  and  minus  forms 
and  where  consequently  there  can,  on  this  score,  be 
no  occasion  for  ambiguity. 

This  last  expression  is  known  as  the  Rule  of  Car- 
Cardan's  dan,  and  there  has  hitherto  been  no  method  devised 
for  the  resolution  of  equations  of  the  third  degree 
which  does  not  lead  to  it.  Since  cubic  radicals  nat- 
urally present  but  a  single  value,  it  was  long  thought 
that  Cardan's  rule  could  give  but  one  of  the  roots  of 
the  equation,  and  that  in  order  to  find  the  two  others 
we  must  have  recourse  to  the  original  equation  and  di- 
vide it  by  x  —  a,  a  being  the  first  root  found.  The 
resulting  quotient  being  an  equation  of  the  second  de- 
gree may  be  resolved  in  the  usual  manner.  The  divi- 
sion in  question  is  not  only  always  possible,  but  it  is 
also  very  easy  to  perform.  For  in  the  case  we  are 
considering  the  equation  being 

X*+pX+g  =  Q, 

if  a  is  one  of  the  roots  we  shall  have 


which  subtracted  from  the  preceding  will  give 


a  quantity  divisible  by  x  —  a  and  having  as  its  result- 

ing quotient 

.v2-f-0JC-f  rt2-f-/  =  0; 

so  that  the  new  equation  which  is  to  be  resolved  for 
finding  the  two  other  roots  will  be 


ON  ALGEBRA.  69 

from  which  we  have  at  once 


I  see  by  the  Algebra  of  Clairaut,  printed  in  1746, 
and  by  D'Alembert's  article  on  the  Irreducible  Case  in  The  gensr- 

ality  of 

the  first  Encyclopaedia  that  the.  idea  referred  to  pre-  algebra, 
vailed  even  in  that  period.  But  it  would  be  the  height 
of  injustice  to  algebra  to  accuse  it  of  not  yielding  re- 
sults which  were  possessed  of  all  the  generality  of 
which  the  question  was  susceptible.  The  sole  re- 
quisite is  to  be  able  to  read  the  peculiar  hand-writing 
of  algebra,  and  we  shall  then  be  able  to  see  in  it  every- 
thing which  by  its  nature  it  can  be  made  to  contain. 
In  the  case  which  we  are  considering  it  was  forgotten 
that  every  cube  root  may  have  three  values,  as  every 
square  root  has  two.  For  the  extraction  of  the  cube 
root  of  a  for  example  is  merely  equivalent  to  the  reso- 
lution of  the  equation  of  the  third  degree  xs  —  a  =  Q. 
Making  x=y$/a,  this  last  equation  passes  into  the 
simpler  form  ys  — 1  =  0,  which  has  the  root  y  =  \. 
Then  dividing  by  y — 1  we  have 


from  which  we  deduce  directly  the  two  other  roots 
—  1  d=  l/— "3 

y=     T 

These  three  roots,  accordingly,  are  the  three  cube 
roots  of  unity,  and  they  may  be  made  to  give  the  three 
cube  roots  of  any  other  quantity  a  by  multiplying 


7O  ON  ALGEBRA. 

them  by  the  ordinary  cube  root  of  that  quantity.  It 
is  the  same  with  roots  of  the  fourth,  the  fifth,  and  all 
the  following  degrees.  For  brevity,  let  us  designate 
the  two  roots 


The  three  -  1  -)-  I/ 3  -1  —  V - 

cube  roots  ^  o  ' 

of  a 

quantity.  by  m  an(j  ^  jj-  w[\\  foe  seen  that  they  are  imaginary, 
although  their  cube  is  real  and  equal  to  1,  as  we  may 
readily  convince  ourselves  by  raising  them  to  the 
third  power.  We  have,  therefore,  for  the  three  cube 
roots  of  a, 

^~a,     m  ^~a,      n  ^'a. 

Now,  in  the  resolution  of  the  equation  of  the  third 
degree  above  considered,  on  coming  to  the  reduced 
expression  ys  =  A,  where  for  brevity  we  suppose 


A=     -|- 
we  deduced  the  following  result  only  : 


But  from  what  we  have  just  seen,  it  is  clear  that  we 
shall  have  not  only 


but  also 

y  =  m^  A       and    y  =  n 
The  root  x  of  the  equation  of  the  third  degree  which 
we  found  equal  to 

y-^ 
}       3/ 

will  therefore  have  the  three  following  values 


ON  ALGEBRA.  Jl 


which  will  be  the  three   roots  of  the  equation  pro-  The  roots 
posed.     But  making  tionsof  the 


third 
gree. 


it  is  clear  that 


whence 


Substituting  *ty ' B  for ~ — -,  and  remarking  that 

3v A 

mn  =  l,  and  that  consequently 

1  1 

m  n 

the  three  roots  which  we  are  considering  will  be  ex- 
pressed as  follows  : 


m 

We  see,  accordingly,  that  when  properly  under- 
stood the  ordinary  method  gives  the  three  roots  di- 
rectly, and  gives  three  only.  I  have  deemed  it  neces- 
sary to  enter  upon  these  slight  details  for  the  reason 
that  if  on  the  one  hand  the  method  was  long  taxed 
with  being  able  to  give  but  one  root,  on  the  other 
hand  when  it  was  seen  that  it  really  gave  three  it  was 
thought  that  it  should  have  given  six,  owing  to  the 


72  ON  ALGEBRA. 

false  employment  of  all  the  possible  combinations  of 
the  three  cubic  roots  of  unity,  viz.,  1,  m,  n,  with  the 
two  cubic  radicals  ~fr  A  and  f/  B. 

We  could  have  arrived  directly  at  the  results  which 
A  direct       we  have  just  found  by  remarking  that  the  two  equa- 

method  of 

reaching         tlOnS 

the  r°°ts.  y*  _|_  23  _|_  ^  _  0 

give 


where  it  will  be  seen  at  once  that  y3  and  zs  are  the 
roots  of  an  equation  of  the  second  degree  of  which 

p'A 

the  second  term  is  q  and  the  third  —  ^=.      This   equa- 

ul 

tion,  which  is  called  the  reduced  equation,  will  accord- 
ingly have  the  form 


and  calling  A  and  B  its  two  roots  we  shall  have  im- 
mediately 

~ 


where  it  will  be  observed  that  A  and  B  have  the  same 
values  that  they  had  in  the  previous  discussion.  Now, 
from  what  has  gone  before,  we  shall  likewise  have 


or      = 


and  the  same  will  also  hold  good  for  z.   But  the  equa- 
tion 


of  which  we  have  employed  the  cube  only,  limits  these 


•    ON  ALGEBRA.  73 

values  and  it  is  easy  to  see  that  the  restriction  requires 
the  three  corresponding  values  of  z  to  be 


whence  follow  for  the  value  of  x,  which  is  equal  to 
y-\-z,  the  same  three  values  which  we  found  above. 

As  to  the  form  of  these  values  it  is  apparent,  first, 
that  so  long  as  A  and  B  are  real  quantities,  one  only  The  form 

e  i  ir  i  •  <-r>i  of  the  roots 

ot  them  can  be  real,  for  m  and  n  are  imaginary.  I  hey 
can  consequently  all  three  be  real  only  in  the  case 
where  the  roots  A  and  B  of  the  reduced  equation  are 
imaginary,  that  is,  when  the  quantity 

£      j 

4  ~*~  27 

beneath  the  radical  sign  is  negative,  which  happens 
only  when  /  is  negative  and  greater  than 


And  this  is  the  so-called  irreducible  case. 
Since  in  this  event 

£_f 

4  ~*~  27 

is  a  negative  quantity,  let  us  suppose  it  equal  to  —  g*, 
g  being  any  real  quantity  whatever.  Then  making, 
for  the  sake  of  simplicity, 


the  two  roots  A  and  B  of  the  reduced  equation  assume 

the  form 

~--T,   B=f—gV~—\. 


74  ON  ALGEBRA.    * 


Now  I  say  that  if      A  -j-      B,  which  is  one  of  the 
The  reality  roots  of  the  equation  of  the  third  degree,  is  real,  then 

of  the  roots 

the  two  other  roots,  expressed  by 


m  n     13     and     n 

will  also  be  real.      Put 


we  shall  have 

t+u  =  h, 

where  h  by  hypothesis  is  a  real  quantity.      Now, 

tu 
therefore 


squaring  the  equation  /-(-  u  =  h  we  have 

/2  +  2/«+«2:=//2; 
from  which  subtracting  4/«  we  obtain 


(/  _  «)2  _  jp  _  4 

I  observe  that  this  quantity  must  necessarily  be  nega- 
tive, for  if  it  were  positive  and   equal  to  &  we  should 

have 

(/__*)«=#, 

whence 

/  —  u  =  k. 
Then  since 

/+»  =  //, 

it  would  follow  that 

h  —  k 


ON  ALGEBRA.  75 

both  of  which  are  real  quantities.      But  then  /3  and  us 
would  also  be  real  quantities,   which   is  contrary  to 
our  hypothesis,  since  these  quantities  are  equal  to  A 
and  B,  both  of  which  are  imaginary. 
The  quantity 


therefore,  is  necessarily  negative.      Let  us  suppose  it 
equal  to  —  £2;  we  shall  have  then 

(/—«)*  =  —#, 
and  extracting  the  square  root 


/  -  U=k\/-  —  1;  The  form 

whence  of  the  two 

cubic  radi- 

h  +  k\/--l_  f/—  h  —  kV—  l_JTg  cals. 

/  —          -  -  -     —  -  V  A,      U  —  —      —  ;=;  —  -V  D. 

Such  necessarily  will  be  the  form  of  the  two  cubic 
radicals 


V-         and 

a  form  at  which  we  can  arrive  directly  by  expanding 
these  roots  according  to  the  Newtonian  theorem  into 
series.  But  since  proofs  by  series  are  apt  to  leave 
some  doubt  in  the  mind,  I  have  sought  to  render  the 
preceding  discussion  entirely  independent  of  them. 
If,  therefore, 

fT+^ffW/, 
we  shall  have 

Ei    and    ?B  = 


Now  we  have  found  above  that 

_!_L  i/ITs  —1  —  1   —3. 

--  ~ 


76  ON  ALGEBRA. 

wherefore,  multiplying  these  quantities  together,  we 
have 

3" 


+  n  &  B  = 
and 


2 

which  are  real  quantities.      Consequently,  if  the  root 
Condition     A  is  real,  the  two  other  roots  also  will  be  real  in  the 

of  the  real-     .....  ...  .  . 

ity  of  the      irreducible  case  and  they  will  be  real  in  that  case  only. 
But  the  invariable  difficulty  is,  to  demonstrate  di- 
rectly that 


which  we  have  supposed  equal  to  h,  is  always  a  real 
quantity  whatever  be  the  values  of  f  and  g.  In  par- 
ticular cases  the  demonstration  can  be  effected  by  the 
extraction  of  the  cube  root,  when  that  is  possible.  For 
example,  if /=2,  £-=11,  we  shall  find  that  the  cube 
root  of  2 -\-  11  V — 1  will  be  2  -j-  I/ — l,and  similarly 
that  the  cube  root  of  2 —  111  — 1  will  be  2  —  V — 1, 
and  the  sum  of  the  radicals  will  be  4.  An  infinite 
number  of  examples  of  this  class  may  be  constructed 
and  it  was  through  the  consideration  of  such  instances 
that  Bombelli  became  convinced  of  the  reality  of  the 
imaginary  expression  in  the  formula  for  the  irreducible 
case.  But  forasmuch  as  the  extraction  of  cube  roots 
is  in  general  possible  only  by  means  of  series,  we  can- 
not arrive  in  this  way  at  a  general  and  direct  demon- 
stration of  the  proposition  under  consideration. 


ON  ALGEBRA.  77 

It  is  otherwise  with  square  roots  and  with  all  roots 
of  which  the  exponents  are  powers  of  2.   For  example,  Extraction 

. ,  ,  .  .  of  the 

it  we  have  the  expression  square 

roots  of  two 

1/V-J-  g\/' 1  _|-  V  f g^ 1  imaginary 

binomials. 

composed  of  two  imaginary  radicals,  its  square  will  be 

a  quantity  which  is  necessarily  positive.  Extracting 
the  square  root,  so  as  to  obtain  the  equivalent  expres- 
sion, we  have 

l/2/+2i/72T?~,  . 

for  the  real  value  of  the  imaginary  quantity  we  started 
with.  But  if  instead  of  the  sum  we  had  had  the  dif- 
ference between  the  two  proposed  imaginary  radicals 
we  should  then  have  obtained  for  its  square  the  fol- 
lowing expression 

a  quantity  which  is  necessarily  negative  ;  and,  taking 
the  square  root  of  the  latter,  we  should  have  obtained 
the  simple  imaginary  expression 


_  2 
Further,  if  the  quantity 


were  given,  we  should  have,  by  squaring,  the  form 


a  real  and  positive  quantity.     Extracting  the  square 


78  ON  ALGEBRA. 

root  of  this  expression  we  should  obtain  a  real  value 
for  the  original  quantity  ;  and  so  on  for  all  the  other 
remaining  even  roots.  But  if  we  should  attempt  to 
apply  the  preceding  method  to  cubic  radicals  we 
should  be  led  again  to  equations  of  the  third  degree 
in  the  irreducible  case. 
For  example,  let 

Extraction  Vf-±g\/~—\  -j-  Vf  --  gV  -^l  =  X. 

of  the  cube 

roots  of  two  Cubing,  we  get 

imaginary  ,     /  -  -  ,/  -  =\ 

binomials.     2/+  3  &f*  +  g2  \Y  f  +  gV  —^  +  ^f~S  V—V  —  X*  ; 

that  is 


or,  with  the  terms  properly  arranged, 


the  general  formula  of  the  irreducible  case,  for 
~ 


~  (2/)2  +       (-  3  f1"  3 


If  g=  0  we  shall  have  x  =  2  f.  The  sole  desideratum, 
therefore,  is  to  demonstrate  that  if  g  have  any  value 
whatever,  x  has  a  corresponding  real  value.  Now  the 
second  last  equation  gives 


3* 
and  cubing  we  get 

'  —  8/3 


27  x3 
whence 

^_Jg9_6jfe/_: 


ON  ALGEBRA.  79 

an  equation  which  may  be  written  as  follows 

^C**  — 8/)(*3+/)» 

27  *a 
or,  better,  thus  : 


27 

It  is  plain  from  the  last  expression  that  g  is  zero 
when  x3  =  8/;  further,    that  g  constantly  and  unin-  General 
terruptedly  increases   as  x  increases  ;  for  the  factor  the°reyaiity 

O3+/)2  augments  constantly,   and  the  other  factor  of  the  roots 

8  / 
1 3  also  keeps  increasing,  seeing  that  as  the  de- 

OC  o  x 

nominator  x3  increases  the  negative  part  -^,  which  is 
originally  equal  to  1,  keeps  constantly  growing  less 
than  1.  Therefore,  if  the  value  of  x3  be  increased  by 
insensible  degrees  from  8/  to  infinity,  the  value  of  g2 
will  also  augment  by  insensible  and  corresponding 
degrees  from  zero  to  infinity.  And  therefore,  recip- 
rocally, to  every  value  of  g*  from  zero  to  infinity  there 
must  correspond  some  value  of  x3  lying  between  the 
limits  of  8/and  infinity,  and  since  this  is  so  whatever 
be  the  value  of  f  we  may  legitimately  conclude  that, 
be  the  values  of  /  and  g  what  they  may,  the  corre- 
sponding value  of  x3  and  consequently  also  of  x  is 
always  real. 

But  how  is  this  value  of  x  to  be  assigned?  It  would 
seem  that  it  can  be  represented  only  by  an  imaginary 
expression  or  by  a  series  which  is  the  development  of 
an  imaginary  expression.  Are  we  to  regard  this  class 
of  imaginary  expressions,  which  correspond  to  real 


8O  ON  ALGEBRA. 

values,  as  constituting  a  new  species  of  algebraical  ex- 

pressions which  although  they  are  not,  like  other  ex*- 

imaginary    pressions,  susceptible  of  being  numerically  evaluated 

expressions    . 

in  the  form  in  which  they  exist,  yet  possess  the  indis- 
putable advantage  —  and  this  is  the  chief  requisite  — 
that  they  can  be  employed  in  the  operations  of  algebra 
exactly  as  if  they  did  not  contain  imaginary  expres- 
sions. They  further  enjoy  the  advantage  of  having  a 
wide  range  of  usefulness  in  geometrical  constructions, 
as  we  shall  see  in  the  theory  of  angular  sections,  so 
that  they  can  always  be  exactly  represented  by  lines  ; 
while  as  to  their  numerical  value,  we  can  always  find 
it  approximately  and  to  any  degree  of  exactness  that 
we  desire,  by  the  approximate  resolution  of  the  equa- 
tion on  which  they  depend,  or  by  the  use  of  the  com- 
mon trigonometrical  tables. 

It  is  demonstrated  in  geometry  that  if  in  a  circle 
having  the  radius  r  an  arc  be  taken  of  which  the  chord 
is  c,  and  that  if  the  chord  of  the  third  part  of  that  arc 
be  called  x,  we  shall  have  for  the  determination  of  x 
the  following  equation  of  the  third  degree 


an  equation  which  leads  to  the  irreducible  case  since 
c  is  always  necessarily  less  than  2r,  and  which,  owing 
to  the  two  undetermined  quantities  r  and  c,  may  be 
taken  as  the  type  of  all  equations  of  this  class.  For, 
if  we  compare  it  with  the  general  equation 


we  shall  have 


ON  ALGEBRA.  8l 


r=^—^and  c  =  -  --± 

so   that  by  trisecting  the  arc  corresponding  to  the 

chord  c  in  a  circle  of  the  radius  r  we  shall  obtain  at  Trisection 

once  the  value  of  a  root  x,  which  will  be  the  chord  of 

the  third  part  of  that  arc.     Now,  from  the  nature  of  a 

circle  the  same  chord  c  corresponds  not  only  to  the 

arc  s  but  (calling  the  entire  circumference  «)  also  to 

the  arcs 

u  —  s,   2u-{-s,   3u — s,  .  .  . 
Also  the  arcs 

u-\-s,  2u — s,  3»-(-j,  .  .  . 

have  the  same  chord,  but  taken  negatively,  for  on 
completing  a  full  circumference  the  chords  become 
zero  and  then  negative,  and  they  do  not  become  posi- 
tive again  until  the  completion  of  the  second  circum- 
ference, as  you'  may  readily  see.  Therefore,  the  val- 
ues of  x  are  not  only  the  chord  of  the  arc-^  but  also 
the  chords  of  the  arcs 

u  —  s      2u-}-  s 

and  these  chords  will  be  the  three  roots  of  the  equa- 
tion proposed.  If  we  were  to  take  the  succeeding  arcs 
which  have  the  same  chord  c  we  should  be  led  simply 
to  the  same  roots,  for  the  arc  3« — s  would  give  the 

chord  of         — ,  that  is,  of  u ^  ,  which  we  have  al- 

d      s 
ready  seen  is  the  same  as  that  oi-^,  and  so  with  the 

rest. 


82  ON  ALGEBRA. 

Since  in  the  irreducible  case  the  coefficient  p  is 

necessarily  negative,  the  value  of  the  given  chord  c 

TriSono-       will  be  positive  or  negative  according  as  q  is  positive 

metrical  so- 

lution.         or  negative.      In   the  first  case,  we  take  for  s  the  arc 

subtended  by  the  positive  chord  c  =  —  —  .      The  sec- 

P 
ond  case  is  reducible  to  the  first  by  making  x  nega- 

tive, whereby  the  sign  of  the  last  term  is  changed  ;  so 
that  if  again  we  take  for  s  an   arc  subtended  by  the 

positive  chord  —  ,   we  shall   have   simply  to   change 

P 
the  sign  of  the  three  roots. 

Although  the  preceding  discussion  may  be  deemed 
sufficient  to  dispel  all  doubts  concerning  the  nature 
of  the  roots  of  equations  of  the  third  degree,  we  pro- 
pose adding  to  it  a  few  reflexions  concerning  the 
method  by  which  the  roots  are  found.  The  method 
which  we  have  propounded  in  the  foregoing  and  which 
is  commonly  called  Cardan's  method,  although  it  seems 
to  me  that  we  owe  it  to  Hudde,  has  been  frequently 
criticised,  and  will  doubtless  always  be  criticised,  for 
giving  the  roots  in  the  irreducible  case  in  an  imaginary 
form,  solely  because  a  supposition  is  here  made  which 
is  contradictory  to  the  nature  of  the  equation.  For 
the  very  gist  of  the  method  consists  in  its  supposing 
the  unknown  quantity  equal  to  two  undetermined 
quantities  y-\-z,  in  .order  to  enable  us  afterwards  to 
separate  the  resulting  equation 


into  the  two  following  : 


ON  ALGEBRA.  83 

0  and/  +  s3  +  ?  =  0. 
Now,  throwing  the  first  of  these  into  the  form 


a    3  _  The  method 

^  o  of  indeter- 


it  is  plain  that  the  question  reduces  itself  to  finding 
two  numbers  j'3  and  z3  of  which  the  sum  is  —  q  and 

P* 
the  product  —  —  ,    which    is    impossible    unless    the 

square  of  half  the  sum  exceed  the  product,  for  the 
difference  between  these  two  quantities  is  equal  to  the 
square  of  half  the  difference  of  the  numbers  sought. 

The  natural  conclusion  was  that  it  was  not  at  all 
astonishing  that  we  should  reach  imaginary  expres- 
sions when  proceeding  from  a  supposition  which  it 
was  impossible  to  express  in  numbers,  and  so  some 
writers  have  been  induced  to  believe  that  by  adopting 
a  different  course  the  expression  in  question  could  be 
avoided  and  the  roots  all  obtained  in  their  real  form. 

Since  pretty  much  the  same  objection  can  be  ad- 
vanced against  the  other  methods  which  have  since 
been  found  and  which  are  all  more  or  less  based  upon 
the  method  of  indeterminates,  that  is,  the  introduc- 
tion of  certain  arbitrary  quantities  to  be  determined 
so  as  to  satisfy  the  conditions  of  the  problem,  —  we 
propose  to  consider  the  question  of  the  reality  of  the 
roots  by  itself  and  independently  of  any  supposition 
whatever.  Let  us  take  again  the  equation 


and  let  us  suppose  that  its  three  roots  are  a,  d,  c. 


84  ON  ALGEBRA. 

By  the  theory  of  equations  the  left-hand  side  of 
the  preceding  expression  is  the  product  of  three  quan- 
tities 

Anindepen-  x  -  a,    X  -  b,    X  -  C, 

dent  con- 

sideration.   which,  multiplied  together,  give 


xs-  —  (a  -f-  b  +  c)  x*  -)-  (a  b  +  a  c  -(-  b  c]  x  —  a  be; 
and  comparing  the  corresponding  terms,  we  have 

a  -{-  &  -\-  <;  =  Q,  ab  -\-  ac  -\-  b  c=p,  abc  =  -  —  q. 
As  the  degree  of  the  equation  is  odd  we  may  be  cer- 
tain, as  you  doubtless  already  know  and  in  any  event 
will  clearly  see  from  the  lecture  which  is  to  follow, 
that  it  has  necessarily  one  real  root.  Let  that  root 
be  c.  The  first  of  the  three  equations  which  we  have 
just  found  will  then  give 

c  =  —  a  —  b, 

whence  it  is  plain  that  a  -f-  b  is  also  necessarily  a  real 
quantity.  Substituting  the  last  value  of  c  in  the  sec- 
ond and  third  equations,  we  have 

ab  —  a1  —  ab  —  ab  —  b'i=p,   —  a  b  (a  -\-  b~)  --=  —  q, 
or 


from  which  are  to  be  found  a  and  b.     The  last  equa- 

tion gives  ab  =  —  ~-  from  which  I  conclude  that  ab 

a-\-  b 

also  is  necessarily  a  real  quantity.      Let  us  consider 

a2        & 
now  the  quantity  ~-  -\-  ~=  or,  clearing  of  fractions,  the 

quantity  27^2-|-4/3,  upon  the  sign  of  which  the  ir- 
reducible case  depends.  Substituting  in  this  for  p 
and  q  their  value  as  given  above  in  terms  of  a  and  b, 


ON   ALGEBRA.  85 

we  shall  find  that  when  the  necessary  reductions  are 
made  the  quantity  in  question  is  equal  to  the  square  of 


Newview 
of  the  real- 


taken   negatively;   so   that  by  changing  the  signs  and  »tyofthe 
extracting  the  square  root  we  shall  have 

2a8  —  2£8-f3a2£  —  3a£2  =  i/  —  270s  —  4/», 

whence  it  is  easy  to  infer  that  the  two  roots  a  and  b 
cannot  be  real  unless  the  quantity  27^2-f  4/>3  be  neg- 
ative. But  I  shall  show  that  in  that  case,  which  is  as 
we  know  the  irreducible  case,  the  two  roots  a  and  b 
are  necessarily  real.  The  quantity 


may  be  reduced  to  the  form 
(a  —  £)(2a2-f 

as  multiplication  will  show.  Now,  we  have  already 
seen  that  the  two  quantities  a-\-  b  and  ab  are  necessa- 
rily real,  whence  it  follows  that 


is  also  necessarily  real.  Hence  the  other  factor  a  —  b 
is  also  real  when  the  radical  I/--  -27^  —  4/3  is  real. 
Therefore  a-\-b  and  a  —  b  being  real  quantities,  it  fol- 
lows that  a  and  b  are  real. 

We  have  already  derived  the  preceding  theorems 
from  the  form  of  the  roots  themselves.  But  the  pres- 
ent demonstration  is  in  some  respects  more  general 
and  more  direct,  being  deduced  from  the  fundamental 
principles  of  the  problem  itself.  We  have  made  no 


86  ON  ALGEBRA. 

suppositions,  and  the  particular  nature  of  the  irredu- 
cible   case   has  introduced   no  imaginary  quantities. 
Final  soiu-          But  the  values  of  a  and  b  still  remain  to  be  found 

tion  on  the 

new  view,     from  the  preceding  equations.     And  to  this  end  I  ob- 
serve that  the  left-hand  side  of  the  equation 

3  1      " 

^  M 

can  be  made  a  perfect  cube  by  adding  the  left-hand 
side  of  the  equation 

a  b  (a  +  fr\  =  q, 


3 
multiplied  by  —         —  ,  and  that  the  root  of  this  cube  is 


so  that,  extracting  the  cube  root  of  both  sides,   we 
shall  have  the  expression 


l_j_3 

—  -  b 


expressed  in  known  quantities.  And  since  the  radical 
I/  —  3  may  also  be  taken  negatively,  we  shall  also 
have  the  expression 


2  2 

expressed  in  known  quantities,  from  which  the  values 
of  a  and  b  can  be  deduced.  And  these  values  will 
contain  the  imaginary  quantity  ]/ — 3,  which  was  in- 
troduced by  multiplication,  and  will  be  reducible  to 
the  same  form  with  the  two  roots 


ON  ALGEBRA.  87 

m  $'A~+  n  I3  B   and  n  f~A  +  m  $/£~, 
which  we  found  above.     The  third  root 

C  =  -  a  -  b  Office  of 

imaginary 

will  then  be  expressed  by  f/^4-\-  ^iB.  quantities 

By  this  method  we  see  that  the  imaginary  quanti- 
ties employed  have  simply  served  to  facilitate  the  ex- 
traction of  the  cube  root  without  which  we  could  not 
determine  separately  the  values  of  a  and  b.  And  since 
it  is  apparently  impossible  to  attain  this  object  by  a 
different  method,  we  may  regard  it  as  a  demonstrated 
truth'  that  the  general  expression  of  the  roots  of  an 
equation  of  the  third  degree  in  the  irreducible  case 
cannot  be  rendered  independent  of  imaginary  quan- 
tities. 

Let  us  now  pass  to  equations  of  the  fourth  degree. 
We  have  already  said  that  the  artifice  which  was  ori- 
ginally employed  for  resolving  these  equations  con- 
sisted in  so  arranging  them  that  the  square  root  of 
the  two  sides  could  be  extracted,  by  which  they  were 
reduced  to  equations  of  the  second  degree.  The  fol- 
lowing is  the  procedure  employed.  Let 


be  the  general  equation  of  the  fourth  degree  deprived 
of  its  second  term,  which  can  always  be  eliminated, 
as  you  know,  by  increasing  or  diminishing  the  roots 
by  a  suitable  quantity.  Let  the  equation  be  put  in 

the  form 

x*  =  —  /  x2  —  q  x  —  /-, 


88  ON  ALGEBRA. 

and  to  each  side  let  there  be  added  the  terms  2x'2y-\- 

y*,  which  contain  a  new  undetermined  quantity  y  but 

Biquadratic  which  still  leave  the  left-hand  side  of  the  equation  a 

equations. 

square.     We  shall  then  have 


We  must  now  make  the  right-hand  side  also  a  square. 
To  this  end  it  is  necessary  that 


in  which  case  the  square  root  of  the  right-hand  side 
will  have  the  form 


2  y  2y—p 

Supposing  then  that  the  quantity  jy  satisfies  the  equa- 
tion 


which  developed  becomes 


, 

2  2         8 

and  which,  as  we  see,  is  an  equation  of  the  third  de- 
gree, the  equation  originally  given  may  be  reduced  to 
the  following  by  extracting  the  square  root  of  its  two 
members,  viz.  : 


—  p 

where  we  may  take  either  the  plus  or  the  positive 
value  for  the  radical  \  2y—p,  and  shall  consequently 
have  two  equations  of  the  second  degree  to  which  the 
given  equation  has  been  reduced  and  the  roots  of 
which  will  give  the  four  roots  of  the  original  equation. 


ON  ALGEBRA.  89 

All  of  which  furnishes  us  with  our  first  instance  of  the 
decomposition  of  equations  into  others  of  lower  de- 
gree. 

The  method  of  Descartes  which  is  commonly  fol- 
lowed in  the  elements  of  algebra  is  based  upon  the  The 

method  of 

same  principle  and  consists  in  assuming  at  the  outset  Descartes. 
that  the  proposed  equation  is  produced  by  the  mul- 
tiplication of  two  equations  of  the  second  degree,  as 

x2  —  ux-\-s  =  Q  and  x*  +  ux  +  t  =  Q, 
where  u,  s,  and  /  are  indeterminate  coefficients.    Mul- 
tiplying them  together  we  have 

*4  +  0+/—  «2)*+0—  f)ux  +  st  =  Q, 
comparison  of  which  with  the  original  equation  gives 

s-\-t  —  u*=£,   (s  —  /)#=r$r  and  st  =  r. 
The  first  two  equations  give 


And  if  these  values  be  substituted  in  the  third  equa- 
tion of  condition  st  =  r,  we  shall  have  an  equation  of 
the  sixth  degree  in  u,  which  owing  to  its  containing 
only  even  powers  of  u  is  resolvable  by  the  rules  for 
cubic  equations.  And  if  we  substitute  in  this  equation 
2y  —  /  for  u2,  we  shall  obtain  in  y  the  same  reduced 
equation  that  we  found  above  by  the  old  method. 

Having  the  value  of  «2  we  have  also  the  values  of 
s  and  /,  and  our  equation  of  the  fourth  degree  will  be 
decomposed  into  two  equations  of  the  second  degree 
which  will  give  the  four  roots  sought.  This  method, 
as  well  as  the  preceding,  has  been  the  occasion  of  some 


90  ON  ALGEBRA. 

hesitancy  as  to  which  of  the  three  roots  of  the  re- 

duced cubic  equation  in  u2  or  y  should  be  employed. 

The  deter-    The  difficulty  has   been  well  resolved    in   Clairaut's 

mined 

character  Algebra,  where  we  are  led  to  see  directly  that  we  al- 
ways obtain  the  same  four  roots  or  values  of  x  what- 
ever root  of  the  reduced  equation  we  employ.  But 
this  generality  is  needless  and  prejudicial  to  the  sim- 
plicity which  is  to  be  desired  in  the  expression  of 
the  roots  of  the  proposed  equation,  and  we  should 
prefer  the  formulae  which  you  have  learned  in  the 
principal  course  and  in  which  the  three  roots  of  the 
reduced  equation  are  contained  in  exactly  the  same 
manner. 

The  following  is  another  method  of  reaching  the 
same  formulae,  less  direct  than  that  which  has  already 
been  expounded  to  you,  but  which,  on  the  other  hand 
has  the  advantage  of  being  analogous  to  the  method 
of  Cardan  for  equations  of  the  third  degree. 

I  take  up  again  the  equation 


and  I  suppose 

x  =jy  -\-z-\-t. 
Squaring  I  obtain 

X1  =y1  _|_  22  _|_  /2  _|_  2  (y  Z 

Squaring  again  I  have 


but 

)2  =yi  z*  +/  /2  +  z'-  P  +  2/  z  t  +  2y  z*  t 
=/  z2  +/2  /2  +  s*  f1  -f  2  y  z  t  (y  +  z  +  /). 


ON  ALGEBRA.  QI 

Substituting  these  three  values  of  x,  x2,  and  _r4  in  the 
original  equation,  and  bringing  together  the  terms 
multiplied  by  y  -\-z-\-t  and  the  terms  multiplied  by  A  third 

method. 

yz-\-yt-\-zt,  I  have  the  transformed  equation 


_j_  s2/2)  +  (  8_j.s/  +  ?)  (7  +  2  +  O  +  r=0. 

We  now  proceed  as  we  did  with  equations  of  the  third 
degree,  where  we  caused  the  terms  containing  y  +  z 
to  vanish,  and  in  the  same  manner  cause  here  the 
terms  containing  j  +  z-\-  t  and  y  z  -\-  y  t  -\-  z.t  to  disap- 
pear, which  will  give  us  the  two  equations  of  condi- 
tion 

=  0    and 


There  remains  the  equation 


and  the  three  together  will  determine  the  quantities 
y,  z,  and  /.     The  second  gives  immediately 


which  substituted  in  the  third  gives 

/22+//2  +  22/.2_^_^ 

The  first,  raised  to«its  square,  gives 

"f=£ 

Hence,  by  the  general  theory  of  equations  the  three 


Q2  ON  ALGEBRA. 

quantities  f2,  z2,  f2  will  be  the  roots  of  an  equation  of 
the  third  degree  having  the  form 

P  I '  P'1        r\  a1 

The  u3  _|_  l_  U2  _|  \u JL.  —  0  J 

reduced  2  \16          4/  64 

equation. 

so  that  if  the  three  roots  of  this  equation,  which  we 
will  call  the  reduced  equation,  be  designated  by  a,  b,  ct 
we  shall  have 

and  the  value  of  x  will  be  expressed  by 

V~a  +  J/T-f  i/TT 

Since  the  three  radicals  may  each  be  taken  with  the 
plus  sign  or  the  minus  sign,  we  should  have,  if  all 
possible  combinations  were  taken,  eight  different  val- 
ues for  x.  It  is  to  be  observed,  however,  that  in  the 

preceding  analysis  we  employed  the  equation  y1  z*- P  = 

a* 

^-p  whereas  the  equation  immediately  given  is  yzt  = 

—  -Q-.      Hence  the  product   of  the  three  quantities  y, 

o 

z,  /,  that  is  to  say  of  the  three  radicals 

y"a,  yJ,  VT, 

must  have  the  contrary  sign  to  that  of  the  quantity  q. 
Therefore,  if  q  be  a  negative  quantity,  either  three 
positive  radicals  or  one  positive  and  two  negative  rad- 
icals must  be  contained  in  the  expression  for  x.  And 
in  this  case  we  shall  have  the  following  four  combina- 
tions only  : 

y'  a  +  V  b  -}-  y  c,      \    a  —  y  b  —  V  c, 


ON  ALGEBRA.  93 

which  will  be  the  four  roots  of  the  proposed  equation 
of  the  fourth  degree.  But  if  q  be  a  positive  quantity, 
either  three  negative  radicals  or  one  negative  and  two  Euier's 

....  .  .          formu4ae. 

positive  radicals  must  be  contained  in  the  expression 
for  x,  which  will  give  the  following  four  other  com- 
binations as  the  roots  of  the  proposed  equation  :* 

-  V~a  —  V~b  —  VT,       -  V~a  +  T/T+  V~c, 
l/H—  1/T+  1/7;     V~a  +  1/7—  V~c. 

Now  if  the  three  roots  a,  b,  c  of  the  reduced  equa- 
tion of  the  third  .degree  are  all  real  and  positive,  it  is 
evident  that  the  four  preceding  roots  will  also  all  be 
real.  But  if  among  the  three  real  roots  a,  b,  c,  any 
are  negative,  obviously  the  four  roots  of  the  given 
biquadratic  equation  will  be  imaginary.  Hence,  be- 
sides the  condition  for  the  reality  of  the  three  roots  of 
the  reduced  equation  it  is  also  requisite  in  the  first 
case,  agreeably  to  the  well-known  rule  of  Descartes, 


*  These  simple  and  elegant  formulas  are  due  to  Euler.  But  M.  Bret,  Pro- 
fessor of  Mathematics  at  Grenoble,  has  made  the  important  observation  (see 
the  Correspondance  sur  I'Ecole  Polytechnique,  t.  II.,  3"ie  Cahier,  p.  217)  that 
they  can  give  false  values  when  imaginary  quantities  occur  among  the  four 
roots. 

In  order  to  remove  all  difficulty  and  ambiguity  we  have  only  to  substitute 
for  one  of  these  radicals  its  value  as  derived  from  the  equation  V  a  V6  \  c  = 
—-.  Then  the  formula 

o 

i/^~    ,     r/   i  _y  

o  I/         1/~Z" 
o  '   tl    '    O 

will  give  the  four  roots  of  the  original  equation  by  taking  for  a  and  b  any  two 
of  the  three  roots  of  the  reduced  equation,  and  by  taking  the  two  radicals 
successively  positive  and  negative. 

The  preceding  remark  should  be  added  to  article  777  of  Euler'i  Algebra 
and  to  article  37  of  the  author's  Note  XIII  of  the  Traiti  de  la  resolution  des 


94  ON  ALGEBRA. 

that  the  coefficients  of  the  terms  of  the  reduced  equa- 
tion should  be  alternatively  positive  and  negative,  and 

/>2        r 
Roots  of  a    consequently  that  p  should  be   negative  and  =^  — 

biquadratic 

equation,  positive,  that  is,  /2  >  4 r.  If  one  of  these  conditions 
is  not  realised  the  proposed  biquadratic  equation  can- 
not have  four  real  roots.  If  the  reduced  equation  have 
but  one  real  root,  it  will  be  observed,  first,  that  by 
reason  of  its  last  term  being  negative  the  one  real  root 
of  the  equation  must  necessarily  be  positive.  It  is 
then  easy  to  see  from  the  general  expressions  which 
we  gave  for  the  roots  of  cubic  equations  deprived  of 
their  second  term, — a  form  to  which  the  reduced  equa- 
tion in  u  can  easily  be  brought  by  simply  increasing 
all  the  roots  by  the  quantity^, — it  is  easy  to  see,  I 
say,  that  the  two  imaginary  roots  of  this  equation  wil! 
be  of  the  form 

/+^i/:ri  and  -f—g i/:rr. 

Therefore,  supposing  a  to  be  the  real  root  and  b,  c  the 
two  imaginary  roots,  1  a  will  be  a  real  quantity  and 
V  b  -j-  V  c  will  also  be  real  for  reasons  which  we  have 
given  above  ;  while  V  b  —  \/  c  on  the  other  hand  will 
be  imaginary.  Whence  it  follows  that  of  the  four 
roots  of  the  proposed  biquadratic  equation,  the  two 
first  will  be  real  and  the  two  others  will  be  imaginary. 
As  for  the  rest,  if  we  make  u  =  s — ~  in  the  re- 
duced equation  in  u,  so  as  to  eliminate  the  second 
term  and  to  reduce  it  to  the  form  which  we  have  above 


ON  ALGEBRA.  95 

It 

examined,   we  shall  have  the   following  transformed 
equation  in  j  : 


-— 

484,          864       24        64  ~ 

and  the  condition  for  the  reality  of  the  three  roots  of 
the  reduced  equation  will  be 


LECTURE  IV. 

ON  THE   RESOLUTION   OF   NUMERICAL   EQUATIONS. 


W 

.imits  of 


E  have  seen  how  equations  of  the  second,  the 
Limits  of        vv     third,  and  the  fourth  degree  can  be  resolved. 
icai  resoiu-  The  fifth  degree  constitutes  a  sort  of  barrier  to  anal- 
ysts, which  by  their  greatest  efforts  they  have  never 

equations.      * 

yet  been  able  to  surmount,  and  the  general  resolution 
of  equations  is  one  of  the  things  that  are  still  to  be 
desired  in  algebra.  I  say  in  algebra,  for  if  with  the 
third  degree  the  analytical  expression  of  the  roots  is 
insufficient  for  determining  in  all  cases  their  numeri- 
cal value,  a  fortiori  must  it  be  so  with  equations  of  a 
higher  degree ;  and  so  we  find  ourselves  constantly 
under  the  necessity  of  having  recourse  to  other  means 
for  determining  numerically  the  roots  of  a  given  equa- 
tion,— for  to  determine  these  roots  is  in  the  last  re- 
sort the  object  of  the  solution  of  all  problems  which 
necessity  or  curiosity  may  offer. 

I  propose  here  to  set  forth  the  principal  artifices 
which  have  been  devised  for  accomplishing  this  im- 
portant object.  Let  us  consider  any  equation  of  the 
»/th  degree,  represented  by  the  formula 


RESOLUTION   OF  NUMERICAL  EQUATIONS.  97 


in  which  x  is  the  unknown  quantity,  /,  q,  r,  .  .  .  .  the 
known    positive   or    negative    coefficients,   and   u  the  conditions 
last  term,  not  containing  x  and  consequently  also  a  J^tion^** 
known  quantity.      It  is   assumed   that   the  values  of  numencal 

equations. 

these  coefficients  are  given  either  in  numbers  or  in 
lines;  (it  is  indifferent  which,  seeing  that  by  taking  a 
given  line  as  the  unit  or  common  measure  of  the  rest 
we  can  assign  to  all  the  lines  numerical  values;)  and  it 
is  clear  that  this  assumption  is  always  permissible 
when  the  equation  is  the  result  of  a  real  and  determi- 
nate problem.  The  problem  set  us  is  to  find  the  value, 
or,  if  there  be  several,  the  values,  of  x  which  satisfy  the 
equation,  i.  e.  which  render  the  sum  of  all  its  terms 
zero.  Now  any  other  value  which  may  be  given  to  x 
will  render  that  sum  equal  to  some  positive  or  nega- 
tive quantity,  for  since  only  integral  powers  of  x  en- 
ter the  equation,  it  is  plain  that  every  real  value  of  x 
will  also  give  a  real  value  for  the  quantity  in  question. 
The  more  that  value  approaches  to  zero,  the  more 
will  the  value  of  x  which  has  produced  it  approach  to 
a  root  of  the  equation.  And  if  we  find  two  values  of 
x,  of  which  one  renders  the  sum  of  the  terms  equal  to 
a  positive  quantity  and  the  other  to  a  negative  quan- 
tity, we  may  be  assured  in  advance  that  between  these 
two  values  there  will  of  necessity  be  at  least  one  value 
which  will  render  the  expression  zero  and  will  con- 
sequently be  a  root  of  the  equation. 

Let  P  stand  for  the   sum   of  all   the   terms  of  the 


98  RESOLUTION  OF  NUMERICAL  EQUATIONS. 

equation  having  the  sign  -)-  and  Q  for  the  sum  of  all 
the  terms  having  the  sign  — ;  then  the  equation  will 
be  represented  by 

P—  Q  =  V. 

Let  us  suppose,  for  further  simplicity,  that  the  two 

Position  of  values  of  x  in  question   are   positive,   that  A  is  the 

smaller,  B  the  greater,  and  that  the  substitution  of  A 

numerical 

equations.  for  x  g{ves  a  negative  result  and  the  substitution  of  B 
for  x  a  positive  result ;  i.  e.,  that  the  value  of  P — Q 
is  negative  when  x  =A,  and  positive  when  x^B. 

Consequently,  when  x  =  A,  P  will  be  less  than  Q, 
and  when  x  =  B,  P  will  be  greater  than  Q.  Now, 
from  the  very  form  of  the  quantities  P  and  Q,  which 
contain  only  positive  terms  and  \vhole  positive  powers 
of  x,  it  is  clear  that  these  quantities  augment  continu- 
ously as  x  augments,  and  that  by  making  x  augment  by 
insensible  degrees  through  all  values  from  A  to  B,  they 
also  will  augment  by  insensible  degrees  but  in  such 
wise  that  P  will  increase  more  than  Q,  seeing  that 
from  having  been  smaller  than  Q  it  will  have  become 
greater.  Therefore,  there  must  of  necessity  be  some 
expression  for  the  value  of  x  between  A  and  B  which 
will  make/*— (?;  just  as  two  moving  bodies  which 
we  suppose  to  be  travelling  along  the  same  straight 
line  and  which  having  started  simultaneously  from 
two  different  points  arrive  simultaneously  at  two  other 
points  but  in  such  wise  that  the  body  which  was  at  first 
in  the  rear  is  now  in  advance  of  the  other, — just  as 
two  such  bodies,  I  say,  must  necessarily  meet  at  some 


RESOLUTION  OF  NUMERICAL  EQUATIONS.  99 

point  in  their  path.  That  value  of  x,  therefore,  which 
will  make  P=  Q  will  be  one  of  the  roots  of  the  equa- 
tion, and  such  a  value  will  lie  of  necessity  between  A 
and  B. 

The    same   reasoning   may   be    employed    for    the  Position  of 
other  cases,  and  always  with  the  same  result.  Lme^cai0 

The  proposition  in  question  is  also  demonstrable  eiuatl 
by  a  direct  consideration  of  the  equation  itself,  which 
may  be  regarded  as  made  up  of  the  product  of  the 

factors, 

_r  —  a,  x  — 1>,  x  —  <:,  .  .  .  .  , 

where  a,  b,  c,  .  .  .  .  are  the  roots.  For  it  is  obvious 
that  this  product  cannot,  by  the  substitution  of  two 
different  values  for  x,  be  made  to  change  its  sign,  un- 
less at  least  one  of  the  factors  changes  its  sign.  And 
it  is  likewise  easy  to  see  that  if  more  than  one  of  the 
factors  changes  its  sign,  their  number  must  be  odd. 
Thus,  if  A  and  B  are  two  values  of  x  for  which  the 
factor  x  —  b,  for  example,  has  opposite  signs,  then  if 
A  be  larger  than  b,  necessarily  B  must  be  smaller 
than  b,  or  vice  versa.  Perforce,  then,  the  root  b  will 
fall  between  the  two  quantities  A  and  B. 

As  for  imaginary  roots,  if  there  be  any  in  the  equa- 
tion, since  it  has  been  demonstrated  that  they  always 
occur  in  pairs  and  are  of  the  form 


therefore  if  a  and  b  are  imaginary,  the  product  of  the 
factors  x — a  and  x  —  b  will  be 


100  RESOLUTION  OF  NUMERICAL  EQUATIONS. 


a  quantity  which  is  always  positive  whatever  value  be 
given  to  x.  From  this  it  follows  that  alterations  in 
the  sign  can  be  due  only  to  real  roots.  But  since  the 
theorem  respecting  the  form  of  imaginary  roots  can- 
not be  rigorously  demonstrated  without  employing  the 
other  theorem  that  every  equation  of  an  odd  degree 
has  necessarily  one  real  root,  a  theorem  of  which  the 
general  demonstration  itself  depends  on  the  proposi- 
tion which  we  are  concerned  in  proving,  it  follows 
that  that  demonstration  must  be  regarded  as  a  sort  of 
vicious  circle,  and  that  it  must  be  replaced  by  another 
which  is  unassailable. 

But  there  is  a  more  general  and  simpler  method 

Application  of  considering  equations,  which  enjoys  the  advantage 

°o  a^bra^  of  affording  direct  demonstration  to  the  eye  of  the 

principal  properties  of  equations.     It  is  founded  upon 

a  species  of  application  of  geometry  to  algebra  which 

is  the  more  deserving  of  exposition  as  it  finds  extended 

employment  in  all  branches  of  mathematics. 

Let  us  take  up  again  the  general  equation  pro- 
posed above  and  let  us  represent  by  straight  lines  all 
the  successive  values  which  are  given  to  the  unknown 
quantity  x  and  let  us  do  the  same  for  the  correspond- 
ing values  which  the  left-hand  side  of  the  equation 
assumes  in  this  manner.  To  this  end,  instead  of  sup- 
posing the  right-hand  side  of  the  equation  equal  to 
zero,  we  suppose  it  equal  to  an  undetermined  quan- 
tity y.  We  lay  off  the  values  of  x  upon  an  indefinite 


RESOLUTION  OF  NUMERICAL  EQUATIONS. 


IOI 


straight  line  AB  (Fig.  1),  starting  from  a  fixed  point 
O  at  which  x  is  zero  and  taking  the  positive  values  of 
x  in  the  direction  OB  to  the  right  of  O  and  the  nega- 
tive values  of  x  in  the  opposite  direction  to  the  left  of 
O.  Then  let  OP  be  any  value  of  x.  To  represent 
the  corresponding  value  of  y  we  erect  at  P  a  perpen- 
dicular to  the  line  OB  and  lay  off  on  it  the  value  of  y 
in  the  direction  PQ  above  the  straight  line  OB  if  it  is 
positive,  and  on  the  same  perpendicular  below  OB  if 
it  is  negative.  We  do  the  same  for  all  the  values  of 


Represen- 
tation of 
equations 
by  curves. 


Fig.  i. 

x,  positive  as  well  as  negative ;  that  is,  we  lay  off 
corresponding  values  of  y  upon  perpendiculars  to  the 
straight  line  through  all  the  points  whose  distance 
from  the  point  O  is  equal  to  x.  The  extremities  of  all 
these  perpendiculars  will  together  form  a  straight  line 
or  a  curve,  which  will  furnish,  so  to  speak,  a  picture 
of  the  equation 

xm -± p xm~l  ^r  q x"1-* -{- .  .  .+u=y. 
The  line  AB  is  called  the  axis  of  the  curve,  O  the  origin 
of  the  abscissae,  OP—x  an  abscissa,  PQ=y  the  cor- 


IO2  RESOLUTION  OF  NUMERICAL  EQUATIONS. 

responding  ordinate,  and  the  equations  in  x  and  y  the 
equations  of  the  curve.  A  curve  such  as  that  of  Fig. 
1  having  been  described  in  the  manner  indicated,  it  is 
clear  that  its  intersections  with  the  axis  AB  will  give 
the  roots  of  the  proposed  equation 

Graphic  x'"  -\- p Xm~l  -\-  q  Xm~'i  -f-  .    .    .  +  U  =  0. 

resolution 

of  equa-       For  seeing  that  this  equation  is  realised  only  when  in 

tions.  . 

the  equation  of  the  curve  y  becomes  zero,  therefore 
those  values  of  x  which  satisfy  the  equation  in  ques- 
tion and  which  are  its  roots  can  only  be  the  abscissae 


that  correspond  to  the  points  at  which  the  ordinates 
are  zero,  that  is,  to  the  points  at  which  the  curve  cuts 
the  axis  AB.  Thus,  supposing  the  curve  of  the  equa- 
tion in  x  and  y  is  that  represented  in  Fig.  1,  the  roots 
of  the  proposed  equation  will  be 

OM,    ON,    OR,   ....   and  —  OI,  —  OG,  .... 
I  give  the  sign  --to  the  latter  because  the  intersec- 
tions /,  G,  .  .  .  fall  on  the  other  side  of  the  point  O. 
The  consideration  of  the  curve  in  question  gives  rise 
to  the  following  general  remarks  upon  equations : 


RESOLUTION  OF  NUMERICAL  EQUATIONS.  103 

(1)  Since  the  equation  of  the  curve  contains  only 
whole  and  positive  powers  of  the  unknown  quantity  x 

it  is  clear  that  to  every  value  of  x  there  must  corre-  Theconse- 
spond  a  determinate  value  of  y,  and  that  the  value  in  the  grlphic 
question  will  be  unique  and  finite  so  long  as  x  is  finite.  resolutlon- 
But  since  there  is  nothing  to  limit  the  values  of  x  they 
may  be  supposed  infinitely  great,  positive  as  well  as 
negative,  and  to  them  will  correspond  also  values  of 
y  which  are  infinitely  great.      Whence  it  follows  that 
the  curve  will  have  a  continuous  and  single  course, 
and  that  it  may  be  extended  to  infinity  on  both  sides 
of  the  origin  O. 

(2)  It  also  follows  that  the  curve  cannot  pass  from 
one  side  of  the  axis  to  the  other  without  cutting  it, 
and  that  it  cannot  return  to  the  same  side  without 
having  cut  it  twice.      Consequently,  between  any  two 
points  of  the  curve  on  the  same  side  of  the  axis  there 
will  necessarily  be  either  no  intersections  or  an  even 
number  of  intersections  ;  for  example,    between   the 
points  H  and   Q  we  find  two  intersections  /  and  M, 
and  between  the  points  //and  S  we  find  four,  /,  M, 
N,  R,  and  so  on.     Contrariwise,  between  a  point  on 
one  side  of  the  axis  and  a  point  on  the  other  side,  the 
curve  will  have  an  odd  number  of  intersections ;  for 
example,  between  the  points  L  and  Q  there  is  one  in- 
tersection M,  and  between  the  points  H  and  K  there 
are  three  intersections,  /,  M,  N,  and  so  on. 

For  the  same  reason  there  can  be  no  simple  inter- 
section unless  on  both  sides  of  the  point  of  intersec- 


IO4  RESOLUTION  OF  NUMERICAL  EQUATIONS. 

tion,  above  and  below  the  axis,  points  of  the  curve  are 

situated  as  are  the  points  L,  Q  with  respect  to  the  in- 

intersec-      tersection  M.      But  two  intersections,  such  as  N  and 

tions  indi- 

catethe  «3  may  approach  each  other  so  as  ultimately  to  coin- 
cide at  T.  Then  the  branch  QKS  will  take  the  form 
of  the  dotted  line  QTS  and  touch  the  axis  at  T,  and 
will  consequently  lie  in  its  whole  extent  above  the 
axis ;  this  is  the  case  in  which  the  two  roots  ON,  OR 
are  equal.  If  three  intersections  coincide  at  a  point, 
— a  coincidence  which  occurs  when  there  are  three 
equal  roots, — then  the  curve  will  cut  the  axis  in  one 
additional  point  only,  as  in  the  case  of  a  single  point 
of  intersection,  and  so  on. 

Consequently,  if  we  have  found  for  y  two  values 
having  the  same  sign,  we  may  be  assured  that  between 
the  two  corresponding  values  of  x  there  can  fall  only 
an  even  number  of  roots  of  the  proposed  equation  ; 
that  is,  that  there  will  be  none  or  there  will  be  two,  or 
there  will  be  four,  etc.  On  the  other  hand,  if  we  have 
found  for  y  two  values  having  contrary  signs,  we  may 
be  assured  that  between  the  corresponding  values  of 
x  there  will  necessarily  fall  an  odd  number  of  roots  of 
the  proposed  equation  ;  that  is,  there  will  be  one,  or 
there  will  be  three,  or  there  will  be  five,  etc. ;  so  that, 
in  the  case  last  mentioned,  we  may  infer  immediately 
that  there  will  be  at  least  one  root  of  the  proposed 
equation  between  the  two  values  of  x. 

Conversely,  every  value  of  x  which  is  a  root  of  the 
equation  will  be  found  between  some  larger  and  some 


RESOLUTION  OF  NUMERICAL  EQUATIONS.  IO5 

smaller  value  of  x  which  on  being  substituted  for  x  in 
the  equation  will  yield  values  of  y  with  contrary  signs. 
This  will  not  be  the  case,  however,  if  the  value  of 
'x  is  a  double  root;  that  is,  if  the  equation  contains  Case  of 
two  roots  of  the  same  value:      On  the  other  hand,  if  roots. 
the  value  of  x  is  a  triple  root,  there  will  again  exist 
a  larger  and  a  smaller  value  for  x  which  will  give  to 
the  corresponding  values  of  y  contrary  signs,  and  so 
on  with  the  rest. 

If,  now,  we  consider  the  equation  of  the  curve,  it 
is  plain  in  the  first  place,  that  by  making  x  =  Q  we 
shall  havej— «/  and  consequently  that  the  sign  of 
the  ordinatejy  will  be  the  same  as  that  of  the  quantity 
u,  the  last  term  of  the  proposed  equation.  It  is  also 
easy  to  see  that  there  can  be  given  to  x  a  positive  or 
negative  value  sufficiently  great  to  make  the  first  term 
x™  of  the  equation  exceed  the  sum  of  all  the  other 
terms  which  have  the  opposite  sign  to  xm ;  with  the 
result  that  the  corresponding  value  of  y  will  have  the 
same  sign  as  the  first  term  x"'.  Now,  if  m  is  odd  xm 
will  be  positive  or  negative  according  as  x  is  positive 
or  negative,  and  if  m  is  even,  xm  will  always  be  posi- 
tive whether  x  be  positive  or  not. 
Whence  we  may  conclude  : 

(1)  That  every  equation  of  an  odd  degree  of  which 
the  last  term  is  negative  has  an  odd  number  of  roots 
between  x  =  Q  and  some  very  large  positive  value  of 
x,  and  an  even  number  of  roots  between  *  =  0  and 
some  very  large  negative  value  of  x,  and  consequently 


106  RESOLUTION  OF   NUMERICAL  EQUATIONS. 

that  it  has  at  least  one  real  positive  root.     That,  con- 
trariwise, if  the  last  term  of  the  equation  is  positive  it 
General       will  have  an  odd  number  of  roots  between  x  =  0  and 
as  to  the       some  very  large   negative  value   of  x,    and   an   even 
number  of  roots  between  a:  =  0  and  some  very  large 

of  the  roots.  * 

positive  value  of  x,  and  consequently  that  it  will  have 
at  least  one  real  negative  root. 

(2)  That  every  equation  of  an  even  degree,  of 
which  the  last  term  is  negative,  has  an  odd  number  of 
roots  between  x  =  0  and  some  very  large  positive  value 
of  x,  as  well  as  an  odd  number  of  roots  between  x  —  Q 
and  some  very  large  negative  value  of  x,  and  conse- 
quently that  it  has  at  least  one  real  positive  root  and 
one  real  negative  root.  That,  on  the  other  hand,  if 
the  last  term  is  positive  there  will  be  an  even  number 
of  roots  between  x  =  Q  and  some  very  large  positive 
value  of  x,  and  also  an  even  number  of  roots  between 
x  =  Q  and  some  very  large  negative  value  of  x  ;  with 
the  result  that  in  this  case  the  equation  may  have  no 
real  root,  whether  positive  or  negative. 

We  have  said  that  there  could  always  be  given  to 
x  a  value  sufficiently  great  to  make  the  first  term  xm  of 
the  equation  exceed  the  sum  of  all  the  terms  of  con- 
trary sign.  Although  this  proposition  is  not  in  need 
of  demonstration,  seeing  that,  since  the  power  xm  is 
higher  than  any  of  the  other  powers  of  x  which  enter 
the  equation,  it  is  bound,  as  x  increases,  to  increase 
much  more  rapidly  than  these  other  powers  ;  never- 
theless, in  order  to  leave  no  doubts  in  the  mind,  we 


RESOLUTION  OF  NUMERICAL  EQUATIONS.  107 

shall  offer  a  very  simple  demonstration  of  it, — a  dem- 
onstration which  will  enjoy  the  collateral  advantage 
of  furnishing  a  limit  beyond  which  we  may  be  certain 
no  root  of  the  equation  can  be  found. 

To  this  end,  let  us  first  suppose  that  x  is  positive, 
and  that  k  is  the  greatest  of  the  coefficients  of  the  Limits  of 
negative  terms.      If  we  make  x-=k-\-\  we  shall  have  rol£rt 

X>n ( k    I     \\m  r=  k  ( k    I     1  V" —  ^     I     ( t>    I     1  \m — 1  equations. 

Similarly, 


(k  +  1)—  »= 

and  so  on  ;  so  that  we  shall  finally  have 


Now  this  quantity  is  evidently  greater  than  the  sum 
of  all  the  negative  terms  of  the  equation  taken  posi- 
tively, on  the  supposition  that  x  =  k-\-  1.  Therefore, 
the  supposition  x-=k-\-  1  necessarily  renders  the  first 
term  xm  greater  than  the  sum  of  all  the  negative  terms. 
Consequently,  the  value  of  y  will  have  the  same  sign 
as  x. 

The  same  reasoning  and  the  same  result  hold  good 
when  x  is  negative.  We  have  here  merely  to  change 
x  into  —  x  in  the  proposed  equation,  in  order  to  change 
the  positive  roots  into  negative  roots,  and  vice  versa. 

In  the  same  way  it  may  be  proved  that  if  any  value 
be  given  to  x  greater  than  k-\-\,  the  value  of  y  will 
still  have  the  same  sign.  From  this  and  from  what 
has  been  developed  above,  it  follows  immediately  that 


108  RESOLUTION  OF  NUMERICAL  EQUATIONS. 

the  equation  can  have  no  root  equal  to  or  greater  than 


live  roots. 


Therefore,  in  general,  if  k  is  the  greatest  of  the 
Limits  of  coefficients  of  the  negative  terms  of  an  equation,  and 
and^'ega™6  ^  ^Y  changing  the  unknown  quantity  x  into  —  x,  k  is 
the  greatest  of  the  coefficients  of  the  negative  terms 
of  the  new  equation,  —  the  first  term  always  being  sup- 
posed positive,  —  then  all  the  real  roots  of  the  equa- 
tion will  necessarily  be  comprised  between  the  limits 

k+l  and—  //  —  I. 

But  if  there  are  several  positive  terms  in  the  equa- 
tion preceding  the  first  negative  term,  we  may  take 
for  k  a  quantity  less  than  the  greatest  negative  coeffi- 
cient. In  fact  it  is  easy  to  see  that  the  formula  given 
above  can  be  put  into  the  form 


+  .    .    .+(*+!)* 

and  similarly  into  the  following 


and  so  on. 

Whence  it  is  easy  to  infer  that  if  m  —  n  is  the  ex- 
ponent of  the  first  negative  term  of  the  proposed  equa- 
tion of  the  mih  degree,  and  if  /  is  the  largest  coeffi- 
cient of  the  negative  terms,  it  will  be  sufficient  if  k  is 
so  determined  that 

/K^+i)«-i  =  /. 

And  since  we  may  take  for  k  any  larger  value  that  we 
please,  it  will  be  sufficient  to  take 


RESOLUTION  OF  NUMERICAL  EQUATIONS.  IOQ 

k"  =--1,  or  k  =  \/  I. 

And  the  same  will  hold  good  for  the  quantity  h  as  the 
limit  of  the  negative  roots. 

If,  now,  the  unknown  quantity  x  be  changed  into 
,  the  largest  roots  of  the  equation  in  x  will  be  con-  superior 

and  infe- 

verted  into  the  smallest  in  the  new  equation  in  z,  and  rior  limits 
conversely.      Having  effected  this  transformation,  and  "jvero^/ 
having  so  arranged  the  terms  according  to  the  powers 
of  z  that  the  first  term  of  the  equation  is  zm,  we  may 
then  in  the  same  manner  seek  for  the  limits  K-\-  1  and 
—  H  —  1  of   the   positive   and    negative   roots    of   the 
equation  in  z. 

Thus  K-\-  1  being  larger  than  the  largest  value  of 

z  or  of—,  therefore,  by  the  nature  of  fractions, 

x  A  -|-  1 

will  be  smaller  than  the  smallest  value  of  x  and  simi- 

larly-—:-      will  be  smaller  than  the  smallest  negative 
H-\-  1 

value  of  x. 

Whence  it  may  be  inferred  that  all  the  positive 
real  roots  will  necessarily  be  comprised  between  the 

limits 


and  that  the  negative  real  roots  will  fall  between  the 
limits 

and  _/;-i. 


There  are  methods  for  finding  still  closer  limits  ; 
but  since  they  require  considerable  labor,  the  preced- 


IIO  RESOLUTION  OF  NUMERICAL  EQUATIONS. 

ing  method  is,  in  the  majority  of  cases,  preferable,  as 
being  more  simple  and  convenient. 

For  example,  if  in  the  proposed  equation  I  -\-  z  be 
A  further      substituted  for  x,   and  if   after  having   arranged   the 

method  for 

finding  the  terms  according  to  the  powers  of  2,  there  be  given  to 
/  a  value  such  that  the  coefficients  of  all  the  terms 
become  positive,  it  is  plain  that  there  will  then  be  no 
positive  value  of  z  that  can  satisfy  the  equation.  The 
equation  will  have  negative  roots  only,  and  conse- 
quently /  will  be  a  quantity  greater  than  the  greatest 
value  of  x.  Now  it  is  easy  to  see  that  these  coeffi- 
cients will  t  Expressed  as  follows  : 

P  +  m  I, 


m  (m  _  1  )(»/  —  2) 


and  so  on.  Accordingly,  it  is  only  necessary  to  seek 
by  trial  the  smallest  value  of  /  which  will  render  them 
all  positive. 

But  in  the  majority  of  cases  it  is  not  sufficient  to 
know  the  limits  of  the  roots  of  an  equation  ;  the  thing 
necessary  is  to  know  the  values  of  those  roots,  at 
least  as  approximately  as  the  conditions  of  the  prob- 
lem require.  For  every  problem  leads  in  its  last  anal- 
ysis to  an  equation  which  contains  its  solution  ;  and 
if  it  is  not  in  our  power  to  resolve  this  equation,  all 


RESOLUTION  OF  NUMERICAL  EQUATIONS.  Ill 

the  pains  expended  upon  its  formulation  are  a  sheer 

loss.      We  may  regard  this  point,   therefore,   as  the 

most  important  in  all  analysis,  and  for  this  reason  I  The  real 

have  felt  constrained  to  make  it  the  principal  subject  fhefindin 

of  the  present  lecture.  of  the  roots. 

From  the  principles  established  above  regarding 
the  nature  of  the  curve  of  which  the  ordinates^  repre- 
sent all  the  values  which  the  left-hand  side  of  an 
equation  assumes,  it  follows  that  if  we  possessed 
some  means  of  describing  this  curve  we  should  obtain 
at  once,  by  its  intersections  with  the  axis,  all  the  roots 
of  the  proposed  equation.  But  for  this  purpose  it  is 
not  necessary  to  have  all  of  the  curve  ;  it  is  sufficient 
to  know  the  parts  which  lie  immediately  above  and 
below  each  point  of  intersection.  Now  it  is  possible 
to  find  as  many  points  of  a  curve  as  we  please,  and  as 
near  to  one  another  as  we  please  by  successively  sub- 
stituting for  x  numbers  which  are  very  little  different 
from  one  another,  but  which  are  still  near  enough  for 
our  purpose,  and  by  taking  for  y  the  results  of  these 
substitutions  in  the  left-hand  side  of  the  equation.  If 
among  the  results  of  these  substitutions  two  be  found 
having  contrary  signs,  we  may  be  certain,  by  the  prin- 
ciples established  above,  that  there  will  be  between 
these  two  values  of  x  at  least  one  real  root.  We  can 
then  by  new  substitutions  bring  these  two  limits  still 
closer  together  and  approach  as  nearly  as  we  wish  to 
the  roots  sought. 

Calling  the  smaller  of  the  two  values  of  x  which 


112  RESOLUTION  OF  NUMERICAL  EQUATIONS. 

have  given  results  with  contrary  signs,  A,  and  the 
larger  B,  and  supposing  that  we  wish  to  find  the 
Separation  value  of  the  root  within  a  degree  of  exactness  denoted 
by  n,  where  n  is  a  fraction  of  any  degree  of  smallness 
we  please,  we  proceed  to  substitute  successively  for  x 
the  following  numbers  in  arithmetical  progression  : 

A  -f  n,   A  +  2n,   A  +  3v, 
or 

B  —  n,  B—2n,   B  —  Zn,   .  .  .  .  , 

until  a  result  is  reached  having  the  contrary  sign  to 
that  obtained  by  the  substitution  of  A  or  of  B.  Then 
one  of  the  two  successive  values  of  x  which  have  given 
results  with  contrary  signs  will  necessarily  be  larger 
than  the  root  sought,  and  the  other  smaller  ;  and  since 
by  hypothesis  these  values  differ  from  one  another 
only  by  the  quantity  n,  it  follows  that  each  of  them 
approaches  to  within  less  than  n  of  the  root  sought, 
and  that  the  error  is  therefore  less  than  n. 

But  how  are  the  initial  values  substituted  for  x  to 
be  determined,  so  a^  on  the  one  hand  to  avoid  as 
many  useless  trials  as  possible,  and  on  the  other  to 
make  us  confident  that  we  have  discovered  by  this 
method  all  the  real  roots  of  this  equation.  If  we  ex- 
amine the  curve  of  the  equation  it  will  be  readily  seen 
that  the  question  resolves  itself  into  so  selecting  the 
values  of  x  that  at  least  one  of  them  shall  fall  between 
two  adjacent  intersections,  which  will  be  necessarily 
the  case  if  the  difference  between  two  consecutive  val- 


RESOLUTION  OF  NUMERICAL  EQUATIONS.  113 

ues  is  less  than  the  smallest  distance  between  two 
adjacent  intersections. 

Thus,  supposing  that  D  is  a  quantity  smaller  than 
the  smallest  distance  between  two  intersections  imme-  TO  find  a 
diately  following  each  other,  we  form  the  arithmetical  less  than 
progression  the  differ- 

ence  be- 

0,    D,    22),     32},    42),     .....    ,  tweenany 

two  roots. 

and  we  select  from  this  progression  only  the  terms 
which  fall  between  the  limits 

'** 


as  determined  by  the  method  already  given.  We  ob- 
tain, in  this  manner,  values  which  on  being  substi- 
tuted for  x  ultimately  give  us  all  the  positive  roots  of 
the  equation,  and  at  the  same  time  give  the  initial 
limits  of  each  root.  In  the  same  manner,  for  obtain- 
ing the  negative  roots  we  form  the  progression 

0,  —D,   —22},  —  3Z>,   —  4£>, 

from  which  we  also  take  only  the  terms  comprised 
between  the  limits 


Thus  this  difficulty  is  resolved.  But  it  still  re- 
mains to  find  the  quantity  D,  —  that  is,  a  quantity 
smaller  than  the  smallest  interval  between  any  two  ad- 
jacent intersections  of  the  curve  with  the  axis.  Since 
the  abscissae  which  correspond  to  the  intersections  are 
the  roots  of  the  proposed  equation,  it  is  clear  that  the 
question  reduces  itself  to  finding  a  quantity  smaller 


114  RESOLUTION  OF  NUMERICAL  EQUATIONS. 

than  the  smallest  difference  between  two  roots,  neg- 
lecting the  signs.  We  have,  therefore,  to  seek,  by  the 
methods  which  were  discussed  in  the  lectures  of  the 
principal  course,  the  equation  whose  roots  are  the  dif- 
ferences between  the  roots  of  the  proposed  equation. 
And  we  must  then  seek,  by  the  methods  expounded 
above,  a  quantity  smaller  than  the  smallest  root  of 
this  last  equation,  and  take  that  quantity  for  the  value 
of  D. 

This  method,  as  we  see,  leaves  nothing  to  be  de- 
Theequa-  sired  as  regards  the  rigorous  solution  of  the  problem, 
ferences1  ^ut  **  labors  under  great  disadvantage  in  requiring 
extremely  long  calculations,  especially  if  the  proposed 
equation  is  at  all  high  in  degree.  For  example,  if  ;;/ 
is  the  degree  of  the  original  equation,  that  of  the  equa- 
tion of  differences  will  be  m(tn  —  1),  because  each  root 
can  be  subtracted  from  all  the  remaining  roots,  the 
number  of  which  is  m  —  1, — which  gives  ;//(///  —  1) 
differences.  But  since  each  difference  can  be  positive 
or  negative,  it  follows  that  the  equation  of  differences 
must  have  the  same  roots  both  in  a  positive  and  in  a 
negative  form;  that  consequently  the  equation  must 
be  wanting  in  all  terms  in  which  the  unknown  quan- 
tity is  raised  to  an  odd  power ;  so  that  by  taking  the 

square  of  the  differences  as  the  unknown  quantity,  this 

m  (m  —  1)  , 
unknown  quantity  can  occur  only  in  the  -  — -th 

degree.  For  an  equation  of  the  mth  degree,  accord- 
ingly, there  is  requisite  at  the  start  a  transformed 


RESOLUTION  OF  NUMERICAL  EQUATIONS.  115 

m(m  —  1)  . 
equation  of  the  -    —  ^  --  -tn  degree,  which  necessitates 

Z 

an  enormous  amount  of  tedious  labor,  if  m  is  at  all 
large.      For  example,  for  an  equation  of  the   10th  de-  impractic- 
gree,  the  transformed  equation  would  be  of  the  45th.  the  method. 
And  since  in  the  majority  of  cases  this  disadvantage 
renders  the  method  almost  impracticable,  it  is  of  great 
importance  to  find  a  means  of  remedying  it. 

To  this  end  let  us  resume  the  proposed  equation  of 
the  mth  degree, 

#"•  4-  p  x»>-i  _|_  q  x™-i  _|_  .  .   .  _(-«  —  0, 

of  which  the  roots  are  a,  b,  c,  .  .  .  .     We  shall  have 
then 


and  also 


Let  b  —  a  =  i.  Substitute  this  value  of  b  in  the  second 
equation,  and  after  developing  the  different  powers  of 
a-\-t  according  to  the  well  known  binomial  theorem, 
arrange  the  resulting  equation  according  to  the  powers 
of  /,  beginning  with  the  lowest.  We  shall  have  the 
transformed  equation 


in  which  the  coefficients  P,  Q,  R,  .  .  .  have  the  follow- 
ing values 


Il6  RESOLUTION  OF  NUMERICAL  EQUATIONS. 


m(m-l)    ,n_ 


-- 


Attempt  to 

remedy  the 

method.  (ill  -  2)  (m  -  3) 


and  so  on.    The  law  of  formation  of  these  expressions 
is  evident. 

Now,  by  the  first  equation  in  a  we  have  P  =  0. 
Rejecting,  therefore,  the  term  P  of  the  equation  in  / 
and  dividing  all  the  remaining  terms  by  /,  the  equa- 
tion in  question  will  be  reduced  to  the  (;//  —  l)th  de- 
gree, and  will  have  the  form 

0  +  ^/+.S/2  +  .  .  . -f-  im-*  =  0. 

This  equation  will  have  for  its  roots  (he  m — 1  dif- 
ferences between  the  root  a  and  the  remaining  roots 
b,  c,  .  .  .  Similarly,  if  b  be  substituted  for  a  in  the  ex- 
pressions for  the  coefficients  Q,  J?,  .  .  .  ,  we  shall  ob- 
tain an  equation  of  which  the  roots  are  the  difference 
between  the  root  b  and  the  remaining  roots  a,  c,  .  .  .  , 
and  so  on. 

Accordingly,  if  a  quantity  can  be  found  smaller 
than  the  smallest  root  of  all  these  equations,  it  will 
possess  the  property  required  and  may  be  taken  for 
the  quantity  D,  the  value  of  which  we  are  seeking. 

If,  by  means  of  the  equation  P=Q,  a  be  eliminated 
from  the  equation  in  /,  we  shall  get  a  new  equation  in 
/  which  will  contain  all  the  other  equations  of  which 
we  have  just  spoken,  and  of  which  it  would  only  be 
necessary  to  seek  the  smallest  root.  But  this  new 


RESOLUTION  OF  NUMERICAL  EQUATIONS.  llj 

equation  in  /  is  nothing  else  than  the  equation  of  dif- 
ferences which  we  sought  to  dispense  with. 

In  the  above  equation  in  /  let  us  put  it  z  —  —  .      We  Further  im- 

provement. 
shall  have  then  the  transformed  equation  in  z, 


and  the  greatest  negative  coefficient  of  this  equation 
will,  from  what  has  been  demonstrated  above,  give  a 
value  greater  than  its  greatest  root  ;  so  that  calling 
L  this  greatest  coefficient,  L  -f-  1  will  be  a  quantity 
greater  than  the  greatest  value  of  z.  Consequently, 

—  -~—  will  be'ta  quantity  smaller  than   the   smallest 

'.  • 

positive  value  of  /;  and  in  like  manner  we  shall  find 

a  quantity  smaller  than  the  smallest  negative  value 
of  /.  Accordingly,  we  may  take  for  D  the  smallest  of 
these  two  quantities,  or  some  quantity  smaller  than 
either  of  them. 

For  a  simpler  result,  and  one  which  is  independent 
of  signs,  we  may  reduce  the  question  to  finding  a 
quantity  L  numerically  greater  than  any  of  the  coeffi- 
cients of  the  equation  in  0,  and  it  is  clear  that  if  we 
find  a  quantity  N  numerically  smaller  than  the  small- 
est value  of  Q  and  a  quantity  M  numerically  greater 

than  the  greatest  value  of  any  of  the  quantities  R, 

M 
S,  .  .  .  ,  we  may  put  L  —  -  •    . 

Let  us  begin  with  finding  the  values  of  M.  It  is 
not  difficult  to  demonstrate,  by  the  principles  estab- 
lished above,  that  if  k  -)-  1  is  the  limit  of  the  positive 


Il8      RESOLUTION  OF  NUMERICAL  EQUATIONS. 

roots  and  —  //  —  1  the  limit  of  the  negative  roots  of 
the  proposed  equation,  and  if  for  a,  k-\-  1  and  —  -h  —  1 
Final  reso-  be  successively  substituted  in  the  expressions  for  R, 
S,  .  .  .  ,  considering  only  the  terms  which  have  the 
same  sign  as  the  first,  —  it  is  easy  to  demonstrate  that 
we  shall  obtain  in  this  manner  quantities  which  are 
greater  than  the  greatest  positive  and  negative  values 
of  7?,  S,  .  .  .  corresponding  to  the  roots  a,  b,  c  .  .  .  of 
the  proposed  equation  ;  so  that  we  may  take  for  M 
the  quantity  which  is  numerically  the  greatest  of 
these. 

It  accordingly  only  remains  to  find  a  value  smaller 
than  the  smallest  value  of  Q.  Now  it  would  seem 
that  we  could  arrive  at  this  in  no  other  way  than  by 
employing  the  equation  of  which  the  different  values 
of  Q  are  the  roots,  —  an  equation  which  can  only  be 
reached  by  eliminating  a  from  the  following  equations: 

a"1  -\-pam~l  -\-  qa'n~'i  4-  .  .  .  -|-«  =  0, 

3  +  .  .  .  =  Q. 


It  can  be  easily  demonstrated  by  the  theory  of 
elimination  that  the  resulting  equation  in  Q  will  be  of 
the  mth  degree,  that  is  to  say,  of  the  same  degree  with 
the  proposed  equation  ;  and  it  can  also  be  demon- 
strated from  the  form  of  the  roots  of  this  equation 
that  its  next  to  the  last  term  will  be  missing.  If,  ac- 
cordingly, we  seek  by  the  method  given  above  a  quan- 
tity numerically  smaller  than  the  smallest  root  of  this 
equation,  the  quantity  found  can  be  taken  for  N.  The 


RESOLUTION  OF  NUMERICAL  EQUATIONS.  1 19 

problem  is  therefore  resolved  by  means  of  an  equation 
of  the  same  degree  as  the  proposed  equation. 

The  upshot  of  the  whole  is  a  follows, — where  for  ReCapim- 
the  sake  of  simplicity  I  retain  the  letter  x  instead  of  Vc 
the  letter  a. 

Let  the  following  be  the  proposed  equation  of  the 
;;/th  degree  : 

xm  -\-pxm~l  -J-  qx"'~'2  -\-  r xm~'A  -j-  .  .  .==0; 
let  k  be  the  largest  coefficient  of  the  negative  terms, 
and  m  —  n  the  exponent  of  x  in  the  first  negative  term. 
Similarly,  let  h  be  the  greatest  coefficient  of  the  terms 
having  a  contrary  sign  to  the  first  term  after  x  has 
been  changed  into — x  ;  and  let  m- — n'  be  the  expo- 
nent of  x  in  the  first  term  having  a  contrary  sign  to 
the  first  term  of  the  equation  as  thus  altered.  Put- 
ting, then, 

f=  \/  k  -{- 1     and     g  =  V h  -j-  1, 

we  shall  have  f  and  — g  for  the  limits  of  the  positive 
and  negative  roots.  These  limits  are  then  substituted 
successively  for  x  in  the  following  formulas,  neglect- 
ing the  terms  which  have  the  same  sign  as  the  first 
term : 

m  (in  —  1)        _,        (in  —  l)(w  —  2) 


'  px* 


_2)(;//_  3) 

' 


— 1)0*  — 2)    __„ 


2.3 

O/_1)O  — 2)O  —  3) 

+  L_  ,  Q  M        4/*--*4-..., 


120  RESOLUTION  OF  NUMERICAL  EQUATIONS. 

and  so  on.   Of  these  formulae  there  will  be  m  —  2.    Let 

the  greatest  of  the  numerical  quantities  obtained  in 

The  arith-    this  manner  be  called  M.    We  then  take  the  equation 

metical 

progression  m  Xm~l  -f-  (/« 1) p Xm~2  -j-  (m 2}gXm~S 

revealing 

the  roots.  +  O —  3)r#*-  4  +  .  .  .=y 

and  eliminate  x  from  it  by  means  of  the  proposed 
equation, — which  gives  an  equation  in  y  of  the  mih 
degree  with  its  next  to  the  last  term  wanting.  Let  V 
be  the  last  term  of  this  equation  injy,  and  T  the  larg- 
est coefficient  of  the  terms  having  the  contrary  sign 
to  V,  supposing  y  positive  as  well  as  negative.  Then 
taking  these  two  quantities  T  and  V  positive,  N  will 
be  determined  by  the  equation 


N         ry 

—  N      \r 


where  n  is  equal  to  the  exponent  of  the  last  term  hav- 
ing the  contrary  sign  to  V.  We  then  take  D  equal  to 
or  smaller  than  the  quantity  ,  ,  and  interpolate 
the  arithmetical  progression  : 

0,  D,  2Z>,  3Z>,  .  .  .  ,  —D,  —  2D,  —  3Z>,  .  .  . 
between  the  limits  f  and  — g.  The  terms  of  these 
progressions  being  successively  substituted  for  x  in 
the  proposed  equation  will  reveal  all  the  real  roots, 
positive  as  well  as  negative,  by  the  changes  of  sign 
in  the  series  of  results  produced  by  these  substitu- 
tions, and  they  will  at  the  same  time  give  the  first 
limits  of  these  roots, — limits  which  can  be  narrowed 
as  much  as  we  please,  as  we  already  know. 


RESOLUTION  OF  NUMERICAL  EQUATIONS.  121 

If  the  last  term  V  of  the  equation  in  y  resulting 
from  the  elimination  of  x  is  zero,  then  ^Vwill  be  zero, 
and  consequently  D  will  be  equal  to  zero.  But  in  Method  of 

,   .  .  .  .  .  .,,    ,  elimination 

this  case  it  is  clear  that  the  equation  in  y  will  have 
one  root  equal  to  zero  and  even  two,  because  its  next 
to  the  last  term  is  wanting.  Consequently  the  equa- 
tion 

mxm-*+(m  —  \')py?H-'*-\-(m—'£)qxm- 3  +  .  .  .=0. 
will  hold  good  at  the  same  time  with  the  proposed 
equation.  These  two  equations  will,  accordingly,  have 
a  common  divisor  which  can  be  found  by  the  ordinary 
method,  and  this  divisor,  put  equal  to  zero,  will  give 
one  or  several  roots  of  the  proposed  equation,  which 
roots  will  be  double  or  multiple,  as  is  easily  apparent 
from  the  preceding  theory ;  for  if  the  last  term  Q  of 
the  equation  in  i  is  zero,  it  follows  that 

/r=:0  and  a  =  b. 

The  equation  in  y  is  reduced,  by  the  vanishing  of  its 
last  term,  to  the  (in  —  2)th  degree, — being  divisible 
by  j2.  If  after  this  division  its  last  term  should  still 
be  zero,  this  would  be  an  indication  that  it  had  more 
than  two  roots  equal  to  zero,  and  so  on.  In  such  a 
contingency  we  should  divide  it  by  y  as  many  times 
as  possible,  and  then  take  its  last  term  for  V,  and  the 
greatest  coefficient  of  the  terms  of  contrary  sign  to  V 
for  T,  in  order  to  obtain  the  value  of  Z>,  which  will 
enable  us  to  find  all  the  remaining  roots  of  the  pro- 
posed equation.  If  the  proposed  equation  is  of  the 
third  degree,  as 


122  RESOLUTION  OF  NUMERICAL  EQUATIONS. 

X3  -(-  q  X  -j-  r  =  0, 

we  shall  get  for  the  equation  in  y, 

/  +  3  qy*  —  4^  —  27  r2  =  0. 
If  the  proposed  equation  is 

x^  -f-  ^  ^c2  -(-  r  x  -\-  s  =  0 
we  shall  obtain  for  the  equation  in  y  the  following  : 


and  so  on. 

Since,  however,  the  finding  of  the  equation  injy  by 
General  tne  ordinary  methods  of  elimination  may  be  fraught 
foramina-  w^^  considerable  difficulty,  I  here  give  the  general 
tion-  formulae  for  the  purpose,  derived  from  the  known 

properties  of  equations.     We  form,  first,  from  the  co- 
efficients/, q,  r  of  the  proposed  equation,  the  quanti 
ties  x\,  x%,  xs,  .  .  .  ,  in  the  following  manner  : 

*i  =  —  /i 

•r2  —  —  px\  —  2g, 

x3  =  —  /  xi  —  q  x\  —  3  r, 


We  then  substitute  in  the  expressions  for  y,  y1,  y*,  .  .  . 
up  to  ym,  after  the  terms  in  x  have  been  developed 
the  quantities  x\  for  x,  x^  for  x*,  x3  for  xs,  and  so  forth, 
and  designate  by  y\,  y^,  }'&, .  .  .  the  values  of  y,  j2,  y3, .  . 
resulting  from  these  substitutions.  We  have  then 
simply  to  form  the  quantities  A,  £,  C from  the  formulae 


RESOLUTION  OF  NUiMERICAL  EQUATIONS.  123 


9  » 


(^  _ 


3 

5 

and  we  shall  have  the  following  equation  in  y: 

The  value,  or  rather  the  limit  of  D,  which  we  find 
by  the  method  just  expounded  may  often  be  much  General 
smaller  than  is  necessary  for  finding  all  the  roots,  but  " 
there  would  be  no  further  inconvenience  in  this  than 
to  increase  the  number  of  successive  substitutions  for 
x  in  the  proposed  equation.  Furthermore,  when  there 
are  as  many  results  found  as  there  are  units  in  the 
highest  exponent  of  the  equation,  we  can  continue 
these  results  as  far  as  we  wish  by  the  simple  addition 
of  the  first,  second,  third  differences,  etc.,  because 
the  differences  of  the  order  corresponding  to  the  de- 
gree of  the  equation  are  always  constant. 

We  have  seen  above  how  the  curve  of  the  proposed 
equation  can  be  constructed  by  successively  giving 
different  values  to  the  abscissas  x  and  taking  for  the 
ordinates  y  the  values  resulting  from  these  substitu- 
tions in  the  left-hand  side  of  the  equation.  But  these 
values  for  y  can  also  be  found  by  another  very  simple 
construction,  which  deserves  to  be  brought  to  your 
notice.  Let  us  represent  the  proposed  equation  by 

=  0 


124 


RESOLUTION   OF  NUMERICAL  EQUATIONS. 


A  second 
construc- 
tion for 
solving 
equations. 


where  the  terms  are  taken  in  the  inverse  order. 
equation  of  the  curve  will  then  be 


The 


Drawing  (Fig.  2)  the  straight  line  OX,  which  we  take 
as  the  axis  of  abscissae  with  O  as  origin,  we  lay  off  on 
this  line  the  segment  OI  equal  to  the  unit  in  terms  of 
which  we  may  suppose  the  quantities  a,  b,  c  .  .  .  ,  to 
be  expressed  ;  and  we  erect  at  the  points  OI  the  per- 

M  T  D 


pendiculars  OD,  IM.     We  then  lay  off  upon  the  line 
OD  the  segments 

OA=a,  AB  =  b,  BC=c,  CD  =  d, 
and  so  on.  Let  OP  =  x,  and  at  the  point  P  let  the 
perpendicular  PTbe  erected.  Suppose,  for  example, 
that  d  is  the  last  of  the  coefficients  a,  o,  c,  .  .  .  ,  so  that 
the  proposed  equation  is  only  of  the  third  degree,  and 
that  the  problem  is  to  find  the  value  of 


The  point  D  being  the  last  of  the  points  determined 
upon  the  perpendicular  OD,  and  the  point  C  the  next 


RESOLUTION  OF  NUMERICAL  EQUATIONS.  125 

to  the  last,  we  draw  through  D  the  line  DM  parallel 
to  the  axis  Of,  and  through  the  point  M  where  this 
line  cuts  the  perpendicular  IM  we  draw  the  straight  The  devei- 

.  optnent  and 

line  CM  connecting  M  with  C.  Then  through  the  solution 
point  6"  where  this  last  straight  line  cuts  the  perpen- 
dicular PT,  we  draw  HSL  parallel  to  Of,  and  through 
the  point  L  where  this  parallel  cuts  the  perpendicular 
IM  we  draw  to  the  point  B  the  straight  line  BL. 
Similarly,  through  the  point  R,  where  this  last  line 
cuts  the  perpendicular  PT,  we  draw  GRK  parallel  to 
Of,  and  through  the  point  K,  where  this  parallel  cuts 
the  perpendicular  IM  we  draw  to  the  first  division 
point  A  of  the  perpendicular  DO  the  straight  line  AK. 
The  point  Q  where  this  straight  line  cuts  the  perpen- 
dicular PT  will  give  the  segment  PQ=y. 

Through  Q  draw  the  line  FQ  parallel  to  the  axis 
OP.     The  two  similar  triangles  CDM  and  CHS  give 


Adding  CB  (/)  we  have 

BH=  c-\-dx. 

Also  the  two  similar  triangles  BHL  and  BGR  give 
HL  (1)  :  HB  (c  +  dx)  =  GR  (*)  :  BG  (  =  c  x  +  dx^. 
Adding  AB  (£)  we  have 


Finally  the  similar  triangles  AGKand  AFQ  give 


=  FQ  (A-)  :FA(=bx  +  ex*  +  dx*), 
and  we  obtain  by  adding  OA  (a) 

OF=  PQ  =  a  +  b  x  +  c  x1  +  dx*  =y. 


126  RESOLUTION  OF  NUMERICAL  EQUATIONS. 

The  same  construction  and  the  same  demonstra- 
tion hold,  whatever  be  the  number  of  terms  in  the 
proposed  equation.  When  negative  coefficients  occur 
among  a,  b,  c,  .  .  .  ,  it  is  simply  necessary  to  take 
them  in  the  opposite  direction  to  that  of  the  positive 
coefficients.  For  example,  if  a  were  negative  we 
should  have  to  lay  off  the  segment  OA  below  the  axis 
OI.  Then  we  should  start  from  the  point  A  and  add 
to  it  the  segment  AB  =  b.  If  b  were  positive,  AB 
would  be  taken  in  the  direction  of  OD;  but  if  b  were 
negative,  AB  would  be  taken  in  the  opposite  direc- 
tion, and  so  on  with  the  rest. 

With  regard  to  x,  OP  is  taken  in  the  direction  of 
OI,  which  is  supposed  to  be  equal  to  positive  unity, 
when  x  is  positive  ;  but  in  the  opposite  direction  when 
x  is  negative. 

It  would  not  be  difficult  to  construct,  on  the  fore- 
A  machine  going  model,  an  instrument  which  would  be  applicable 
nations'5  *°  a^  values  °f  the  coefficients  a,  b,  c,  .  .  .  ,  and  which 
by  means  of  a  number  of  movable  and  properly  jointed 
rulers  would  give  for  every  point  P  of  the  straight 
line  OP  the  corresponding  point  Q,  and  which  could 
be  even  made  by  a  continuous  movement  to  describe 
the  curve.  Such  an  instrument  might  be  used  for 
solving  equations  of  all  degrees;  at  least  it  could  be 
used  for  rinding  the  first  approximate  values  of  the 
roots,  by  means  of  which  afterwards  more  exact  values 
could  be  reached. 


LECTURE  V. 

ON  THE  EMPLOYMENT  OF  CURVES  IN  THE  SOLUTION 
OF  PROBLEMS. 

AS  LONG  as  algebra  and  geometry  travelled  sep- 
*^-  arate  paths  their  advance  was  slow  and  their  Geometry 
applications  limited.  But  when  these  two  sciences  aj^ra 
joined  company,  they  drew  from  each  other  fresh  vi- 
tality and  thenceforward  marched  on  at  a  rapid  pace 
towards  perfection.  It  is  to  Descartes  that  we  owe 
the  application  of  algebra  to  geometry, — an  applica- 
tion which  has  furnished  the  key  to  the  greatest  dis- 
coveries in  all  branches  of  mathematics.  The  method 
which  I  last  expounded  to  you  for  finding  and  demon- 
strating divers  general  properties  of  equations  by  con- 
sidering the  curves  which  represent  them,  is,  properly 
speaking,  a  species  of  application  of  geometry  to  al- 
gebra, and  since  this  method  has  extended  applica- 
cations,  and  is  capable  of  readily  solving  problems 
whose  direct  solution  would  be  extremely  difficult  or 
even  impossible,  I  deem  it  proper  to  engage  your  at- 
tention in  this  lecture  with  a  further  view  of  this  sub- 


128  THE  EMPLOYMENT  OF  CURVES. 

ject, — especially  since   it   is   not  ordinarily  found  in 
elementary  works  on  algebra. 

You  have  seen  how  an  equation  of  any  degree 
Method  of  whatsoever  can  be  resolved  by  means  of  a  curve,  of 
by  curves,  which  the  absciss®  represent  the  unknown  quantity 
of  the  equation,  and  the  ordinates  the  values  which 
the  left-hand  member  assumes  for  every  value  of  the 
unknown  quantity.  It  is  clear  that  this  method  can  be 
applied  generally  to  all  equations,  whatever  their  form, 
and  that  it  only  requires  them  to  be  developed  and 
arranged  according  to  the  different  powers  of  the  un- 
known quantity.  It  is  simply  necessary  to  bring  all 
the  terms  of  the  equation  to  one  side,  so  that  the  other 
side  shall  be  equal  to  zero.  Then  taking  the  unknown 
quantity  for  the  abscissa  x,  and  the  function  of  the 
unknown  quantity,  or  the  quantity  compounded  of 
that  quantity  and  the  known  quantities,  which  forms 
one  side  of  the  equation,  for  the  ordinatej',  the  curve 
described  by  these  co-ordinates  x  and  y  will  give  by 
its  intersections  with  the  axis  those  values  of  x  which 
are  the  required  roots  of  the  equation.  And  since 
most  frequently  it  is  not  necessary  to  know  all  pos- 
sible values  of  the  unknown  quantity  but  only  such  as 
solve  the  problem  in  hand,  it  will  be  sufficient  to  de- 
scribe that  portion  of  the  curve  which  corresponds  to 
these  roots,  thus  saving  much  unnecessary  calculation. 
We  can  even  determine  in  this  manner,  from  the  shape 
of  the  curve  itself,  whether  the  problem  has  possible 
solutions  satisfying  the  proposed  conditions. 


THE  EMPLOYMENT  OF  CURVES.  I  2Q 

Suppose,  for  instance,  that  it  is  required  to  find  on 
the  line  joining  two  luminous  points  of  given  intensity, 
the  point  which  receives  a  given  quantity  of  light, —  problem  of 
the  law  of  physics  being  that  the  intensity  of  light  de-  jigeh|™' 
creases  with  the  square  of  the  distance. 

Let  a  be  the  distance  between  the  two  lights  and 
x  the  distance  between  the  point  sought  and  one  of 
the  lights,  the  intensity  of  which  at  unit  distance  is 

M,  the  intensity  of  the  other  at  that  distance  being 

M  N 

N.     The    expressions      ..   and  -  — ~,   accordingly, 

x2  (a  —  x)2 

give  the  intensity  of  the  two  lights  at  the  point  in 
question,  so  that,  designating  the  total  given  effect  by 
A,  we  have  the  equation 

M  N 

-^  -f  :— — ^  =  A 

or 

M  N 


We  will  now  consider  the  curve  having  the  equa- 
tion 

M  N 

I A  y 

in  which  it  will  be  seen  at  once  that  by  giving  to  x  a 

M 
very  small  value,  positive  or  negative,  the  term  — £- , 

X 

while  continuing  positive,  will  grow  very  large,  be- 
cause a  fraction  increases  in  proportion  as  its  denomi- 
nator decreases,  and  it  will  be  infinite  when  #  —  0. 

M 
Further,  if  x  be  made  to  increase,  the  expression     2 

will   constantly  diminish;    but   the   other  expression 


130  THE   EMPLOYMENT  OF  CURVES. 

N  N 

5,  which  was  --.  when  ^  =  0,  will  constantly  m- 
1 


(a  —  *)2 

crease  until  it  becomes  very  large  or  infinite  when  x 

has  a  value  very  near  to  or  equal  to  a. 

Accordingly,  if,  by  giving  to  x  values  from  zero  to 
various  a,  the  sum  of  these  two  expressions  can  be  made  to 
become  less  than  the  given  quantity  A,  then  the  value 
of  y,  which  at  first  was  very  large  and  positive,  will 
become  negative,  and  afterwards  again  become  very 
large  and  positive.  Consequently,  the  curve  will  cut 
the  axis  twice  between  the  two  lights,  and  the  prob- 
lem will  have  two  solutions.  These  two  solutions  will 
be  reduced  to  a  single  solution  if  the  smallest  value  of 

M  N 

~  9  \9 


is  exactly  equal  to  A,  and  they  will  become  imaginary 
if  that  value  is  greater  than  A,  because  then  the  value 
of  y  will  always  be  positive  from  x  =  Q  to  x  =  a. 
Whence  it  is  plain  that  if  one  of  the  conditions  of  the 
problem  be  that  the  required  point  shall  fall  between 
the  two  lights  it  is  possible  that  the  problem  has  no 
solution.  But  if  the  point  be  allowed  to  fall  on  the 
prolongation  of  the  line  joining  the  two  lights,  we 
shall  see  that  the  problem  is  always  resolvable  in  two 
ways.  In  fact,  supposing  x  negative,  it  is  plain  that 
the  term  -y  will  always  remain  positive  and  from  being 
very  large  when  x  is  near  to  zero,  it  will  commence 
and  keep  decreasing  as  x  increases  until  it  grows  very 
small  or  becomes  zero  when  x  is  very  great  or  infinite. 


THE   EMPLOYMENT  OF  CURVES.  131 

N 
The   other   term  ^          -  ,  which  at  first  was  equal  to 

//  (a~xY 

-3-,  also   goes   on   diminishing  until  it  becomes  zero 

0 

when  x  is  negative  infinity.      It  will  be  the  same  if  x 

is  positive   and   greater  than  a  ;  for  when  x  =  a,  the 

N 

expression  -  -  -  will  be  infinitely  great  :  afterwards 

(a  —  x}* 

it  will  keep  on  decreasing  until  it  becomes  zero  when  x 

M 

is  infinite,  while  the  other  expression  —  =  will    first    be 

M  x 

equal  to  —2  and  will  also  go  on  diminishing  towards 

zero  as  x  increases. 

Hence,  whatever  be  the  value  of  the  quantity  A, 
it   is  plain  that  the  values  of  y  will  necessarily  pass  General 
from  positive  to  negative,  both  for  x  negative  and  for  sc 
x   positive   and  greater  than  a.     Accordingly,   there 
will  be  a  negative  value  of  x  and  a  positive  value  of  x 
greater  than  a  which  will  resolve  the  problem  in  all 
cases.      These  values  may  be   found  by  the  general 
method  by  successively  causing  the  values  of  x  which 
give  values  of  y  with   contrary  signs,    to    approach 
nearer  and  nearer  to  each  other. 

With  regard  to  the  values  of  x  which  are  less  than 
a  we  have  seen  that  the  reality  of  these  values  de- 
pends on  the  smallest  value  of  the  quantity 

M  AT 


Directions  for  finding  the  smallest  and  greatest  values 
of  variable  quantities  are  given  in  the  Differential  Cal- 
culus. We  shall  here  content  ourselves  with  remark- 


Minimal 
values. 


132  THE   EMPLOYMENT  OF  CURVES. 

ing  that  the  quantity  in  question  will  be  a  minimum 
when 


3  I  •"* 
a  -  x  S~    \   W 


so  that  we  shall  have 


from  which  we  get,  as  the  smallest  value  of  the  ex- 
pression 


M  N 

9         I        / 


the  quantity 


a 

Hence  there  will  be  two  real  values  for  x  if  this  quan- 
tity is  less  than  A  ;  but  these  values  will  be  imaginary 
if  it  is  greater.  The  case  of  equality  will  give  two 
equal  values  for  x. 

I  have  dwelt  at  considerable  length  on  the  analysis 
of  this  problem,  (though  in  itself  it  is  of  slight  im- 
portance,) for  the  reason  that  it  can  be  made  to  serve 
as  a  type  for  all  analogous  cases. 

The  equation  of  the  foregoing  problem,  having 
been  freed  from  fractions,  will  assume  the  following 
form  : 

Ax*  (a  —  xf  —  M(a  —  x)*  —  Nx*  =  Q. 

With  its  terms  developed  and  properly  arranged  it 
will  be  found  to  be  of  the  fourth  degree,  and  will  con- 
sequently have  four  roots.  Now  by  the  analysis  which 
we  have  just  given,  we  can  recognise  at  once  the  char- 


THE  EMPLOYMENT  OF  CURVES.  133 

acter  of  these  roots.  And  since  a  method  may  spring 
from  this  consideration  applicable  to  all  equations  of 
the  fourth  degree,  we  shall  make  a  few  brief  remarks  Preceding 

.     .  .  _  analysis  ap- 

upon  it  in  passing.      JLet  the  general  equation  be  piied  to  bi- 

4.     i      ,      9     I  ,  quadratic 

*4+/**  +  ^+r  =  0.  equations. 

We  have  already  seen  that  if  the  last  term  of  this 
equation  be  negative  it  will  necessarily  have  two  real 
roots,  one  positive  and  one  negative  ;  but  that  if  the 
last  term  be  positive  we  can  in  general  infer  nothing 
as  to  the  character  of  its  roots.  If  we  give  to  this 
equation  the  following  form 

(^_-aS)»4-£(;r+«)»-f-*(#--tf)*e==a, 
a  form  which  developed  becomes 
#4  _|_  (j  _|_  ^_2<z2)  a,-2  +  2a  (b  —  c}  x  -\-  a*  +  <z2  (6  +  c)  =  0, 

and  from  this  by  comparison  derive  the  following 
equations  of  condition 


and  from  these,  again,  the  following, 

b  +  c=p  +  1ai,    b  —  c=^-,   304  +  /a2  = 
we  shall  obtain,  by  resolving  the  last  equation, 


P  _L     L  +  j. 

-  G  "  \!  3 '"  36- 

If  r  be  supposed  positive,  a2  will  be  positive  and  real, 
and  consequently  a  will  be  real,  and  therefore,  also, 
b  and  c  will  be  real. 

Having  determined  in  this  manner  the  three  quan- 
tities a,  b,  c,  we  obtain  the  transformed  equation 

( *2  _  a2)2  +  b  (X  +  a?  +  C  (X  —  flf  =  0. 


134  THE  EMPLOYMENT  OF  CURVES. 

Putting  the  right-hand  side  of  this  equation  equal 

to  y,  and  considering  the  curve  having  for  abscissae 

Considera-    the  different  values  of  y,  it  is  plain,  that  when  b  and 

tion  of  .   .  ....  -Hi- 

equations  c  are  positive  quantities  this  curve  will  he  wholly 
fourth  de-  ab°ve  the  axis  and  that  consequently  the  equation 
gree.  W.QJ  have  no  real  root.  Secondly,  suppose  that  b  is  a 

negative  quantity  and  c  a  positive  quantity;  then  x  =  a 
will  give  y  =  ±ba1,  —  a  negative  quantity.  A  very 
large  positive  or  negative  x  will  then  give  a  very  large 
positive  y, — whence  it  is  easy  to  conclude  that  the 
equation  will  have  two  real  roots,  one  larger  than  a 
and  one  less  than  a.  We  shall  likewise  find  that  if 
b  is  positive  and  c  is  negative,  the  equation  will  have 
two  real  roots,  one  greater  and  one  less  than  — a. 
Finally,  if  b  and  c  are  both  negative,  then  y  will  be- 
come negative  by  making 

x^a  and  x= — a 

and  it  will  be  positive  and  very  large  for  a  very  large 
positive  or  negative  value  of  x, — whence  it  follows 
that  the  equation  will  have  two  real  roots,  one  greater 
than  a  and  one  less  than  — a.  The  preceding  consid- 
erations might  be  greatly  extended,  but  at  present  we 
must  forego  their  pursuit. 

It  will  be  seen  from  the  preceding  example  that 
the  consideration  of  the  curve  does  not  require  the 
equation  to  be  freed  from  fractional  expressions.  The 
same  may  be  said  of  radical  expressions.  There  is 
an  advantage  even  in  retaining  these  expressions  in 


ves. 


THE  EMPLOYMENT  OF  CURVES.  135 

the  form  given  by  the  analysis  of  the  problem  ;  the 
advantage  being  that  we  may  in  this  way  restrict  our 
attention  to  those  signs  of  the  radicals  which  answer  Advantages 
to  the  special  exigencies  of  each  problem,  instead  of  method  of 
causing  the  fractions  and  the  radicals  to  disappear  cur 
and  obtaining  an  equation  arranged  according  to  the 
different  whole  powers  of  the  unknown  quantity  in 
which  frequently  roots  are  introduced  which  are  en- 
tirely foreign  to  the  question  proposed.  It  is  true  that 
these  roots  are  always  part  of  the  question  viewed  in 
its  entire  extent ;  but  this  wealth  of  algebraical  analy- 
sis, although  in  itself  and  from  a  general  point  of  view 
extremely  valuable,  may  be  inconvenient  and  burden- 
some in  particular  cases  where  the  solution  of  which 
we  are  in  need  cannot  by  direct  methods  be  found  in- 
dependently of  all  other  possible  solutions.  When 
the  equation  which  immediately  flows  from  the  condi- 
tions of  the  problem  contains  radicals  which  are  essen- 
tially ambiguous  in  sign,  the  curve  of  that  equation 
(constructed  by  making  the  side  which  is  equal  to 
zero,  equal  to  the  ordinatejy)  will  necessarily  have  as 
many  branches  as  there  are  possible  different  combi- 
nations of  these  signs,  and  for  the  complete  solution  it 
would  be  necessary  to  consider  each  of  these  branches. 
But  this  generality  may  be  restricted  by  the  particular 
conditions  of  the  problem  which  determine  the  branch 
on  which  the  solution  is  to  be  sought  ;  the  result  being 
that  we  are  spared  much  needless  calculation,  —  an 
advantage  which  is  not  the  least  of  those  offered  by 


136          THE  EMPLOYMENT  OF  CURVES. 

the  method  of  solving  equations  from  the  considera- 
tion of  curves. 

But  this   method  can  be  still  further  generalised 
The  curve    and  even  rendered  independent  of  the  equation  of  the 

of  errors.  .  .  „     .  .  ...  .  . 

problem.  It  is  sufficient  in  applying  it  to  consider 
the  conditions  of  the  problem  in  and  for  themselves, 
to  give  to  the  unknown  quantity  different  arbitrary 
values,  and  to  determine  by  calculation  or  construc- 
tion the  errors  which  result  from  such  suppositions 
according  to  the  original  conditions.  Taking  these 
errors  as  the  ordinates  y  of  a  curve  having  for  abscissae 
the  corresponding  values  of  the  unknown  quantity, 
we  obtain  a  continuous  curve  called  the  curve  of  errors, 
which  by  its  intersections  with  the  axis  also  gives  all 
solutions  of  the  problem.  Thus,  if  two  successive  er- 
rors be  found,  one  of  which  is  an  excess,  and  another 
a  defect,  that  is,  one  positive  and  one  negative,  we 
may  conclude  at  once  that  between  these  two  corre- 
sponding values  of  the  unknown  quantity  there  will 
be  one  for  which  the  error  is  zero,  and  to  which  we 
can  approach  as  near  as  we  please  by  successive  sub- 
stitutions, or  by  the  mechanical  description  of  the 
curve. 

This  mode  of  resolving  questions  by  curves  of  er- 
rors is  one  of  the  most  useful  that  have  been  devised. 
It  is  constantly  employed  in  astronomy  when  direct 
solutions  are  difficult  or  impossible.  It  can  be  em- 
ployed for  resolving  important  problems  of  geometry 
and  mechanics  and  even  of  physics.  It  is  properly 


THE  EMPLOYMENT  OF  CURVES. 


137 


speaking  the  regula  falsi,  taken  in  its  most  general 
sense  and  rendered  applicable  to  all  questions  •uhere 
there  is  an  unknown  quantity  to  be  determined.  It  Solution  of 

a  problem 

can  also  be  applied  to  problems  that  depend  on  two  bythe 
or  several  unknown  quantities  by  successively  giving 
to  these  unknown  quantities  different  arbitrary  values 
and  calculating  the  errors  which  result  therefrom,  af- 
terwards linking  them  together  by  different  curves,  or 
reducing  them  to  tables  ;  the  result  being  that  we  may 
by  this  method  obtain  directly  the  solution  sought 


without  preliminary  elimination  of  the  unknown  quan- 
tities. 

We  shall  illustrate  its  use  by  a  few  examples. 

Required  a  circle  in  which  a  polygon  of  given  sides  can 
be  inscribed. 

This  problem  gives  an  equation  which  is  propor- 
tionate in  degree  to  the  number  of  sides  of  the  poly- 
gon. To  solve  it  by  the  method  just  expounded  we 
describe  any  circle  ABCD  (Fig.  3)  and  lay  off  in  this 
circle  the  given  sides  AB,  BC,  CD,  DE,  EF  of  the 


138  THE  EMPLOYMENT  OF  CURVES. 

polygon,  which  for  the  sake  of  simplicity  I  here  sup- 
pose to  be  pentagonal.      If  the  extremity  of  the  last 
problem  of  side  falls  on  A,  the  problem   is  solved.      But  since  it 
and  in-        is  very  improbable  that  this  should  happen  at  the  first 

?yCg0ned  P°~  trial  we  lay  off  on  the  stra'ght  line  PR  (Fig-  4)  the 
radius  PA  of  the  circle,  and  erect  on  it  at  the  point 
A  the  perpendicular  AF  equal  to  the  chord  AF  of  the 
arc  AF  which  represents  the  error  in  the  supposition 
made  regarding  the  length  of  the  radius  PA.  Since 
this  error  is  an  excess,  it  will  be  necessary  to  describe 


RA' 


Fig.  4. 

a  circle  having  a  larger  radius  and  to  perform  the 
same  operation  as  before,  and  so  on,  trying  circles  of 
various  sizes.  Thus,  the  circle  having  the  radius  PA 
gives  the  error  F'A'  which,  since  it  falls  on  the  hither 
side  of  the  point  A',  should  be  accounted  negative.  It 
will  consequently  be  necessary  in  Fig.  4  in  applying 
the  ordinate  A'F'  to  the  abscissa  PA'  to  draw  that 
ordinate  below  the  axis.  In  this  manner  we  shall  ob- 
tain several  points  F,  F' ,  .  .  .  ,  which  will  lie  on  a 
curve  of  which  the  intersection  R  with  the  axis  PA 


errors. 


THE    EMPLOYMENT  OF  CURVES.  139 

will  give  the  true  radius  PR  of  the  circle  satisfying 
the  problem,   and  we   shall  find  this  intersection  by 
successively  causing  the  points  of  the  curve  lying  on  solution  or 
the   two  sides  of  the  axis  as  F,  F' .  .  .   to  approach  problem  by 
nearer  and  nearer  to  one  another. 

From  a  point,  the  position  of  which  is  unknown,  three 
objects  are  observed,  the  distances  of  which  from  one  an- 
other are  known.  The  three  angles  formed  by  the  rays  of 
light  from  these  three  objects  to  the  eye  of  the  observer  are 
also  known.  Required  the  position  of  the  observer  with 
respect  to  the  three  objects. 

If  the  three  objects  be  joined  by  three  straight 
lines,  it  is  plain  that  these  three  lines  will  form  with 
the  visual  rays  from  the  eye  of  the  observer  a  triangu- 
lar pyramid  of  which  the  base  and  the  three  face  an- 
gles forming  the  solid  angle  at  the  vertex  are  given. 
And  since  the  observer  is  supposed  to  be  stationed  at 
the  vertex,  the  question  is  accordingly  reduced  to  de- 
termining the  dimensions  of  this  pyramid. 

Since  the  position  of  a  point  in  space  is  completely 
determined  by  its  three  distances  from  three  given 
points,  it  is  clear  that  the  problem  will  be  resolved,  if 
the  distances  of  the  point  at  which  the  observer  is 
stationed  from  each  of  the  three  objects  can  be  deter- 
mined. Taking  these  three  distances  as  the  unknown 
quantities  we  shall  have  three  equations  of  the  second 
degree,  which  after  elimination  will  give  a  resultant 
equation  of  the  eighth  degree  ;  but  taking  only  one  of 
these  distances  and  the  relations  of  the  two  others  to  it 


140 


THE    EMPLOYMENT  OF  CURVES. 


for  the  unknown  quantities,  the  final  equation  will  be 

only  of  the  fourth  degree.     We  can  accordingly  rigor- 

Probiem  of  ously  solve  this  problem  by  the  known  methods ;  but 

the  ob-  .  .... 

server  and  the  direct  solution,  which  is  complicated  and  incon- 
venient in  practice,  may  be  replaced  by  the  following 
which  is  reached  by  the  curve  of  errors. 

Let  the  three  successive  angles  APB,  BPC,  CPD 
(Fig.  5)  be  constructed,  having  the  vertex  P  and 
respectively  equal  to  the  angles  observed  between  the 
first  object  and  the  second,  the  second  and  the  third, 


the  third  and  the  first ;  and  let  the  straight  line  PA 
be  taken  at  random  to  represent  the  distance  from  the 
observer  to  the  first  object.  Since  the  distance  of 
that  object  to  the  second  is  supposed  to  be  known, 
let  it  be  denoted  by  AB,  and  let  it  be  laid  off  on  the 
line  AB.  We  shall  in  this  way  obtain  the  distance 
BP  of  the  second  object  to  the  observer.  In  like  man- 
ner, let  BC,  the  distance  of  the  second  object  to  the 
third,  be  laid  off  on  BC,  and  we  shall  have  the  dis- 
tance PC  of  that  object  to  the  observer.  If,  now,  the 


THE   EMPLOYMENT  OF  CURVES.  14! 

distance  of  the  third  object  to  the  first  be  laid  off  on 
the  line  CD,  we  shall  obtain  PD  as  the  distance  of 
the  first  object  to  the  observer.  Consequently,  if  the  Empioy- 

. .  ,,  ,    .  ment  of  the 

distance  first  assumed  is  .exact,  the  two  lines  PA  and  curve  of 
PD  will  necessarily  coincide.      Making,  therefore,  on  61 
the    line  PA,    prolonged    if   necessary,    the    segment 
PE  =  PD,  if  the  point  E  does  not  fall  upon  the  point 
A,  the  difference  will  be  the  error  of  the  first  assump- 
tion PA.    Having  drawn  the  straight  line  PR  (Fig.  6) 
we  lay  off  upon  it  from  the  fixed  point  P,  the  abscissa 
PA,  and  apply  to  it  at  right  angles  the  ordinate  EA  ; 
we  shall  have  the  point  E  of  the  curve  of  errors  ERS. 


Fig  6. 

Taking  other  distances  for  PA,  and  making  the  same 
construction,  we  shall  obtain  other  errors  which  can  be 
similarly  applied  to  the  line  PR,  and  which  will  give 
other  points  in  the  same  curve. 

We  can  thus  trace  this  curve  through  several 
points,  and  the  point  R  where  it  cuts  the  axis  PR  will 
give  the  distance  PR,  of  which  the  error  is  zero,  and 
which  will  consequently  represent  the  exact  distance 
of  the  observer  from  the  first  object.  This  distance 
being  known,  the  others  may  be  obtained  by  the  same 
construction. 

It  is  well  to  remark  that  the  construction  we  have 
been  considering  gives  for  each  point  A  of  the  line 


1^2  THE  EMPLOYMENT  OF  CURVES. 

PA,  two  points  B  and  B'  of  the  line  PB;  for,  since 

the  distance  AB  is  given,  to  find  the  point  B  it  is  only 

Eight  pos-    necessary  to  describe  from  the  point  A  as  centre  and 

sible  solu-          •  t  *'  j  n  <•  •       i  •  -i 

tionsof  the  with  radius  AB  an  arc  of  a  circle  cutting  the  straight 
preceding     H      pB         he  points  B  and  B',—  both  of  which 

problem. 

points  satisfy  the  conditions  of  the  problem.  In  the 
same  manner,  each  of  these  last-mentioned  points  will 
give  two  more  upon  the  straight  line  PC,  and  each  of 
the  last  will  give  two  more  on  the  straight  line  PD. 
Whence  it  follows  that  every  point  A  taken  upon  the 
straight  line  PA  will  in  general  give  eight  upon  the 
straight  line  PD,  all  of  which  must  be  separately  and 
successively  considered  to  obtain  all  the  possible  so- 
lutions. I  have  said,  /#  general,  because  it  is  possible 
(1)  for  the  two  points  B  and  B'  to  coincide  at  a  single 
point,  which  will  happen  when  the  circle  described 
with  the  centre  A  and  radius  AB  touches  the  straight 
line  PB;  and  (2)  that  the  circle  may  not  cut  the 
straight  line  PB  at  all,  in  which  case  the  rest  of  the 
construction  is  impossible,  and  the  same  is  also  to  be 
said  regarding  the  points  C,  D.  Accordingly,  drawing 
the  line  GF  parallel  to  BP  and  at  a  distance  from  it 
equal  to  the  given  line  AB,  the  point  F  at  which  this 
line  cuts  the  line  PE,  prolonged  if  necessary,  will  be 
the  limit  beyond  which  the  points  A  must  not  be  taken 
if  we  desire  to  obtain  possible  solutions.  There  exist 
also  limits  for  the  points  B  and  C,  which  may  be  em- 
ployed in  restricting  the  primitive  suppositions  made 
with  respect  to  the  distance  PA. 


THE  EMPLOYMENT  OF  CURVES.  143 

The  eight  points  D,  which  depend  in  general  on 
each  point  A,  answer  to  the  eight  solutions  of  which 
the  problem  is  susceptible,  and  when  one  has  no  spe-  Reduction 
cial  datum  by  means  of  which  it  can  be  determined  sibieloh^ 
which  of  these  solutions  answer  best  to  the  case  pro    tlonsin 

practice. 

posed,  it  is  indispensable  to  ascertain  them  all  by  em- 
ploying for  each  one  of  the  eight  combinations  a  spe- 
cial curve  of  errors.  But  if  it  be  known,  for  example, 
that  the  distance  of  the  observer  to  the  second  object 
is  greater  or  less  than  his  distance  to  the  first,  it  will 
then  be  necessary  to  take  on  the  line  PB  only  the 
point  B  in  the  first  case  and  the  point  B'  in  the  sec- 
ond,— a  course  which  will  reduce  the  eight  combina- 
tions one-half.  If  we  had  the  same  datum  with  regard 
to  the  third  object  relatively  to  the  second,  and  with 
regard  to  the  first  object  relatively  to  the  third,  then 
the  points  C  and  D  would  be  determined,  and  we 
should  have  but  a  single  solution. 

These  two  examples  may  suffice  to  illustrate  the 
uses  to  which  the  method  of  curves  can  be  put  in  solv- 
ing problems.  But  this  method,  which  we  have  pre- 
sented, so  to  speak,  in  a  mechanical  manner,  can  also 
be  submitted  to  analysis. 

The  entire  question  in  fact  is  reducible  to  the  de- 
scription of  a  curve  which  shall  pass  through  a  certain 
number  of  points,  whether  these  points  be  given  by 
calculation  or  construction,  or  whether  they  be  given 
by  observation  or  single  experiences  entirely  inde- 
pendent of  one  another.  The  problem  is  in  truth  in- 


144  THE  EMPLOYMENT  OF  CURVES. 

determinate,  for  strictly  speaking  there  can  be  made 

to  pass  through  a  given  number  of  points  an  infinite 

General       number  of  different  curves,  regular  or  irregular,  that 

conclusion 

on  the          is,   subject  to  equations  or  arbitrarily  drawn  by  the 
curves         hand.      But  the  question  is  not  to  find  any  solutions 
whatever  but  the  simplest  and  easiest  in  practice. 

Thus  if  there  are  only  two  points  given,  the  sim- 
plest solution  is  a  straight  line  between  the  two  points. 
If  there  are  three  points  given,  the  arc  of  a  circle  is 
drawn  through  these  points,  for  the  arc  of  a  circle 
after  the  straight  line  is  the  simplest  line  that  can  be 
described. 

But  if  the  circle  is  the  simplest  curve  with  respect 
to  description,  it  is  not  so  with  respect  to  the  equa- 
tion between  its  abscissae  and  rectangular  ordinates. 
In  this  latter  point  of  view,  those  curves  may  be  re- 
garded as  the  simplest  of  which  the  ordinates  are  ex- 
pressed by  an  integral  rational  function  of  the  ab- 
scissae, as  in  the  following  equation 


where  y  is  the  ordinate  and  x  the  abscissa.  Curves 
of  this  class  are  called  in  general  parabolic,  because 
they  may  be  regarded  as  a  generalisation  of  the  para- 
bola, —  a  curve  represented  by  the  foregoing  equation 
when  it  has  only  the  first  three  terms.  We  have  al- 
ready illustrated  their  employment  in  resolving  equa- 
tions, and  their  consideration  is  always  useful  in  the 
approximate  description  of  curves,  for  the  reason  that 
a  curve  of  this  kind  can  always  be  made  to  pass 


THE  EMPLOYMENT  OF  CURVES.  145 

through  as  many  points  of  a  given  curve  as  we  please, 
— it  being  only  necessary  to  take  as  many  undeter- 
mined coefficients  a,  b,  r,  ...  as  there  are  points  given,  Parabolic 

•  n-     •  curves. 

and  to  determine  these  coefficients  so  as  to  obtain  the 
abscissae  and  ordinates  for  these  points.  Now  it  is 
clear  that  whatever  be  the  curve  proposed,  the  para- 
bolic curve  so  described  will  always  differ  from  it  by 
less  and  less  according  as  the  number  of  the  different 
points  is  larger  and  larger  and  their  distance  from 
one  another  smaller  and  smaller. 

Newton  was  the  first  to  propose  this  problem.  The 
following  is  the  solution  which  he  gave  of  it  : 

Let  P,  Q,  R,  S,  .  .  .  .  be  the  values  of  the  ordi- 
nates y  corresponding  to  the  values  /,  q,  r,  s,  .  .  .  of 
the  abscissae  x  ;  we  shall  have  the  following  equations 


The  number  of  these  equations  must  be  equal  to  the 
number  of  the  undetermined  coefficients  a,  b,  c,  .  .  .  . 
Subtracting  these  equations  from  one  another,  the  re- 
mainders will  be  divisible  by  q — /,  r — q,  .  .  .  ,  and 
we  shall  have  after  such  division 
Q  —  P 

7^7" 

£=1-2=*  4 

r-q    ~ 


146  THE  EMPLOYMENT  OF  CURVES. 

Let 

Newton's      We  shall  find  in  like  manner,  by  subtraction  and  di- 

problem.  .    .  ,        r    n 

vision,  the  following  : 


r  —  p 
s  —  q 

Further  let 
Ai—Qi 

We  shall  have 


and  so  on. 

In  this  manner  we  shall  find  the  value  of  the  co- 
efficients a,  b,  c,  .  .  .  commencing  with  the  last  ;    and, 
substituting  them  in  the  general  equation 
y  =  a  -\-  b  x  -j-  cot?  -(-  dxA  -f-  .  .  ., 

we  shall  obtain,  after  the  appropriate  reductions  have 
been  made,  the  formula 


which  can  be  carried  as  far  as  we  please. 

But  this  solution  may  be  simplified  by  the  follow- 
ing consideration. 

Since  y  necessarily  becomes  P,  Q,  R  .  .  .  ,  when  x 


THE   EMPLOYMENT  OF  CURVES.  147 

becomes  /,  q,  r,  it  is  easy  to  see  that  the  expression 
for  y  will  be  of  the  form 

y=AP+3Q+  CR  +  DS+  .......  (2)  si 


tion  of 

where  the  quantities  A,  B,  C,  .  .  .  are  so  expressed  in  Newton's 
terms  of  x  that  by  making  x=p  we  shall  have 

A  =  l,   £  =  0,    C=0,  .  .  ., 
and  by  making  x  =  q  we  shall  have 

4  =  0,   B=\,    C=Q,   D  =  G,  .  .  ., 
and  by  making  x  =  r  we  shall  similarly  have 
A  =  Q,   £  =  0,    C=l,   D  =  Q,  .  .  .  etc. 
Whence  it  is  easy  to  conclude  that  the  values  of  A, 
B,  C,  .  .  .  must  be  of  the  form 

.  _  (x  —  q)(x  —  r)(x  —  s).  .  .  . 
~ 


r)(f 


where  there  are  as  many  factors  in  the  numerators 
and  denominators  as  there  are  points  given  of  the 
curve  less  one. 

The  last  expression  for  y  (see  equation  2),  although 
different  in  form,  is  the  same  as  equation  1.  To  show 
this,  the  values  of  the  quantities  Q\,  RI,  63,  ...  need 
only  be  developed  and  substituted  in  equation  1  and 
the  terms  arranged  with  respect  to  the  quantities  P, 
Q,  Jt,  .  .  .  But  the  last  expression  for  y  (equation  2) 
is  preferable,  partly  because  of  the  simplicity  of  the 


148  THE  EMPLOYMENT  OF  CURVES. 

analysis  from  which  it  is  derived,  and  also  because  of 
its  form,  which  is  more  convenient  for  computation. 
Possible  Now,  by  means  of  this  formula,  which  it  is  not 

Newton's  difficult  to  reduce  to  a  geometrical  construction,  we 
are  able  to  find  the  value  of  the  ordinate  y  for  any  ab- 
scissa x,  because  the  ordinates  P,  Q,  R,  .  .  .  for  the 
given  abscissae  p,  q,  r,  .  .  .  are  known.  Thus,  if  we 
have  several  of  the  terms  of  any  series,  we  can  find 
any  intermediate  term  that  we  wish, — an  expedient 
which  is  extremely  valuable  for  supplying  lacunas 
which  may  arise  in  a  series  of  observations  or  experi- 
ments, or  in  tables  calculated  by  formulae  or  in  given 
constructions. 

If  this  theory  now  be  applied  to  the  two  examples 
discussed  above  and  to  similar  examples  in  which  we 
have  errors  corresponding  to  different  suppositions,  we 
can  directly  find  the  error  y  which  corresponds  to  any 
intermediate  supposition  x  by  taking  the  quantities 
P,  Q,  J?,  .  .  .  ,  for  the  errors  found,  and  /,  q,  r,  .  .  .  for 
the  suppositions  from  which  they  result.  But  since 
in  these  examples  the  question  is  to  find  not  the  error 
which  corresponds  to  a  given  supposition,  but  the 
supposition  for  which  the  error  is  zero,  it  is  clear  that 
the  present  question  is  the  opposite  of  the  preceding 
and  that  it  can  also  be  resolved  by  the  same  formula 
by  reciprocally  taking  the  quantities  p,  q,  r,  .  .  .  for 
the  errors,  and  the  quantities  P,  Q,  R,  .  .  .  for  the 
corresponding  suppositions.  Then  x  will  be  the  error 
•for  the  supposition  y;  and  consequently,  by  making 


THE   EMPLOYMENT  OF  CURVES.  149 

x  =  Q,  the  value  of  y  will  be  that  of  the  supposition 
for  which  the  error  is  zero. 

Let  P,  Q,  R,  .  .  .  be  the  values  of  the  unknown 
quantity  in  the  different  suppositions,  and  p,  q,  r  .  .  .  Application 

.    .  i-i     of  Newton's 

the  errors  resulting  from  these  suppositions,  to  which  problem  to 
the  appropriate  signs  are  given.     We  shall  then  have  \*  £[*"_ 
for  the  value  of  the  unknown  quantity  of  which  the  Ples- 
error  is  zero,  the  expression 

AP  +  BQ+CR  + 

in  which  the  values  of  A,  B,  C  .  .  .  are 


p  —  r        q  — 

where  as  many  factors  are  taken  as  there  are  supposi- 
tions less  one. 


APPENDIX. 


NOTE  ON  THE  ORIGIN  OF  ALGEBRA. 


THE  impression  (p.  54)  that  Diophantus  was  the 
"inventor"  of  algebra,  which  sprang,  in  its  Dio- 
phantine  form,  full-fledged  from  his  brain,  was  a  wide- 
spread one  in  the  eighteenth  and  in  the  beginning  of 
the  nineteenth  century.  But,  apart  from  the  intrinsic 
improbability  of  this  view  which  is  at  variance  with 
the  truth  that  science  is  nearly  always  gradual  and 
organic  in  growth,  modern  historical  researches  have 
traced  the  germs  and  beginnings  of  algebra  to  a  much 
remoter  date,  even  in  the  line  of  European  historical 
continuity.  The  Egyptian  book  of  Ahmes  contains 
examples  of  equations  of  the  first  degree.  The  early 
Greek  mathematicians  performed  the  partial  reso- 
lution of  equations  of  the  second  and  third  degree 
by  geometrical  methods.  According  to  Tannery,  an 
embryonic  indeterminate  analysis  existed  in  Pre- 
Christian  times  (Archimedes,  Hero,  Hypsicles).  But 
the  merit  of  Diophantus  as  organiser  and  inaugu- 
rator  of  a  more  systematic  short-hand  notation,  at 
least  in  the  European  line,  remains ;  he  enriched 
whatever  was  handed  down  to  him  with  the  most 
manifold  extensions  and  applications,  betokening  his 


152  APPENDIX. 

originality  and  genius,  and  carried  the  science  of  al- 
gebra to  its  highest  pitch  of  perfection  among  the 
Greeks.  (See  Cantor,  Geschichte  der  Mathematik,  sec- 
ond edition,  Vol.  I.,  p.  438,  et  seq. ;  Ball,  Short  Ac- 
count of  the  History  of  Mathematics,  second  edition,  p. 
104  et  seq.;  Fink,  A  Brief  History  of  Mathematics,  pp. 
63  et  seq.,  77  et  seq.  (Chicago:  The  Open  Court 
Publishing  Co.) 

The  development  of  Hindu  algebra  is  also  to  be 
noted  in  connexion  with  the  text  of  pp.  59-60.  The 
Arabs,  who  had  considerable  commerce  with  India, 
drew  not  a  little  of  their  early  knowledge  from  the 
works  of  the  Hindus.  Their  algebra  rested  on  both 
that  of  the  Hindus  and  the  Greeks.  (See  Ball,  op. 
cit.,  p.  150  et  seq.;  Cantor,  op.  cit.,  Vol.  I.,  p.  651  et 
seq.).  —  Trans. 


INDEX. 


Academies,  rise  of,  62,  63. 

Ahmes,  151. 

Algebra,  definition  of,  2;  history  of, 
54  et  seq.,  151;  essence  of,  55;  the 
name  of,  59;  among  the  Arabs,  59 
et  seq,  152;  in  Europe,  60;  in  Italy, 
64;  in  India,  152;  the  generality  of, 
69;  hand-writing  of,  69;  application 
of  geometry  to,  100  et  seq  ,  127  et 
seq. 

Algebraical  resolution  of  equations, 
limits  of  the,  96. 

Alligation,  generally,  44  et  seq.;  al- 
ternate, 47. 

Analysis,   indeterminate,  47  et  seq., 

55- 

Angle,  trisection  of  an,  62,  81. 

Angular  sections,  theory  of,  80. 

Annuities,   16. 

Apollonius,  54,  59. 

Arabs,  Algebra  among  the,  59  et  seq., 
152. 

Archimedes,  54,  58  footnote,  151. 

Arithmetic,  universal,  2  et  seq. ;  ope- 
rations of,  24  et  seq. 

Arithmetical  progression  revealing 
the  roots,  112  et  seq.,  120. 

Arithmetical  proportion.  12. 

Astronomy,  mechanics,  and  physics, 
curves  of  errors  in,  136. 

Average  life,  45  et  seq. 

Bachet  de  Meziriac,  58. 

Ball,  152. 

Binomial  theorem,  115. 

Binomials,  extraction  of  the  square 

roots  of  two  imaginary,  77. 
Biquadratic  equations,  63,  88,  94,  133. 


Bombelli,  63,  64. 
Bret,  M.,  93  footnote. 
Briggs,  20. 
Buteo,  61. 

Cantor,   54  footnote  ;  60,  footnote, 
152- 

Cardan,  60,  61,  68,  82,  90. 

Checks  on  multiplication  and  divi- 
sion, 39. 

Circle,  144  ;  squaring  of  the,  62;  and 
inscribed  polygon,  problem  of  the, 
138. 

Clairaut,  69,  90. 

Coefficients,  indeterminate,  89;  great- 
est negative,  107  et  seq.,  117. 

Common  divisor  of  two  equations, 
121. 

Complements,  subtraction  by,  26. 

Constantinople,  58. 

Continued  fractions,  solution  of  alli- 
gation by,  50  et  seq. 

Convergents,  7. 

Cube,  duplication  of  the,  62. 

Cube  roots  of  a  quantity,  the  three, 
70. 

Cubic  radicals,  75. 

Curves,  representation  of  equations 
by,  101  et  seq  ;  employment  of  in 
the  solution  of  problems,  127-149; 
method  of,  submitted  to  analysis, 
143  et  seq.  ;  advantages  of  the 
method  of,  135,  144. 

Decimal,  fractions,  9 ;  numbers,  27  et 
seq. 

Decimals,  multiplication  of,  30;  di- 
vision of,  31. 


154 


INDEX. 


DeMorgan,  v. 

Descartes,  viii,  60,  65,  89,  93,  127. 

Differences,   the  equation  of,   114  et 

seq.,  123. 

Differential  Calculus,  131. 
Diophantine  problems,  55. 
Diophantus,  54  et  seq,  151. 
Division,  by  nine,  34  ;  by  eight,  34  ;  by 

seven,  34  et  seq.;  of  decimals,  31. 
Divisor,  greatest  common,  2  et  seq. 
Diihring,  E.  v. 
Duodecimal  system,  32. 

Ecole  Normale,  v,  xi,  12. 

Economy  of  thought,  vii. 

Efflux,  law  of,  42. 

Eleven,  the  number,  test  of  divisibil- 
ity by,  37. 

Elimination,  method  of,  121 ;  general 
formulae  for,  122. 

Equations,  of  the  second  degree,  56; 
of  the  third  degree,  60,  66,  82 ;  of 
ths  fourth  degree,  63,  87,  133;  of 
the  fifth  degree,  64  ;  theory  of,  65, 
84;  biquadratic,  88;  limits  of  the 
algebraical  resolution  of,  96;  of 
the  filth  degree,  96  ;  of  the  »zth  de- 
gree. 96;  general  remarks  upon  the 
roots  of,  102  et  seq.;  graphic  reso- 
lution of,  102;  of  an  odd  degree, 
roots  of,  105  ;  of  an  even  degree, 
roots  of,  106;  real  roots  of,  limits 
of  the,  107  et  seq.;  common  divisor 
of  two,  121 ;  constructions  for  solv- 
ing, 100  et  seq.,  124;  a  machine  for 
solving,  126. 

Equi-different  numbers,  13. 

Errors,  curve  of,  136  et  seq. 

Euclid,  2,  57. 

Euler,  viii,  x,  93. 

Europe,  algebra  in  60. 

Evolution,  ii,  40. 

Experiments,  average  of,  46;  an. ex- 
pedient for  supplying  lacunae  in  a 
series  of,  148. 

Falling  stone,  spaces  traversed  by  a, 

42. 

False,  rule  of,  137. 
Fermat,  58. 
Ferrari,  Louis,  64. 


Ferrous,  Scipio,  60  et  seq. 

Fifth  degree,  equations  of  the,  96. 

Fink,  152. 

Fourth  degree,  equations  of  the,  133. 

Fractional   expressions  in  equations, 

134- 
Fractions,  2  et  seq.;  continued,  3  et 

seq.;    converging,    6;    decimal,   9; 

origin  of  continued,  10. 
France,  58,  61. 

Galileo,  ix. 

Geometers,  ancient,  54  et  seq.,  58,  59. 

Geometrical,  proportion,  13;  calcu- 
lus, 24. 

Geometry,  24,  60;  application  of  to 
algebra,  100  et  seq.,  127  et  seq. 

Germany,  61. 

Girard,  Albert,  62. 

Grain,  of  different  prices,  44. 

Greeks,  mathematics  of  the,  vii,  54  et 
seq.,  151. 

Hand-writing  of  algebra,  69. 

Harriot,  65. 

Hero,  59,  151. 

Horses,  43. 

Hudde,  65,  82. 

Huygens,  ix,  10. 

Hypsicles,  151. 

Imaginary   binomials,    square    roots 

of,  77. 

Imaginary  expressions,  79  et  seq.,  83. 
Imaginary  quantities,   office   of   the, 

87. 

Imaginary  roots,  occur  in  pairs,  99. 
Indeterminate  analysis,  47  et  seq.,  55. 
Indeterminate  coefficients,  89. 
Indeterminates,  the  method  of,  83. 
Ingredients,  48. 
Interest,  15, 
Intersections,   with   the  axis    give 

roots,  102  et  seq  ,  113. 
Inventors,  great,  22. 
Involution  and  evolution,  u. 
Irreducible  case,  61,  65,  69,  73,  82. 
Italy,  cradle  of  algebra  in  Europe, 

61,  64. 

Laborers,  work  of,  41. 
Lagrange,  J.  L.,  v,  vii  et  seq. 


INDEX. 


'55 


Laplace,  v,  xi. 

Lavoisier,  xii. 

Leibnitz,  viii. 

Life  insurance,  45  et  seq 

Life,  probability  of,  46. 

Light,  law  of  the  intensity  of,  129. 

Lights,  problem  of  the  two,  129  et 
seq. 

Limits  of  roots,  107-120. 

Logarithms,  16  et  seq.,  40  ;  advan- 
tages in  calculating  by,  28;  origin 
of,  19;  tables  of,  20. 

Machine  for  solving  equations,  124- 
126. 

Mathematics,  wings  of,  24;  exactness 
of,  43;  evolution  of,  vii. 

Mean  values,  45  et  seq. 

Mechanics,  astronomy,  and  physics, 
curves  of  errors  in,  136. 

Metals,  mingling  of,  by  fusion,  44. 

Meziriac,  Bachet  de,  58. 

Minimal  values,  132. 

Mixtures,  rule  of,  44  et  seq.,  49. 

Monge,  v,  xi. 

Mortality,  tables  of,  45. 

Moving  bodies,  two,  98. 

Multiple  roots,  105. 

Multiplication,  abridged  methods  of, 
26  et  seq.;  inverted,  28;  approxi- 
mate, 29;  of  decimals,  30. 

Music,  22. 

Napier,  17  et  seq. 

Napoleon,  xii. 

Negative  roots,  60. 

Newton,  his  problem,  145;  viii. 

Nine,  property  of  the  number,  31  et 
seq.;  property  of  the  number  gen- 
eralised, 33. 

Nizze,  58  footnote. 

Numeration,  systems  of,  i. 

Numerical  equations,  resolution  of, 
96-126;  conditions  of  the  resolution 
of,  97  ;  position  of  the  roots  of,  98. 
See  Equations; 

Observations,  expedient  for  supply- 
ing lacuna;  in  series  of,  148. 

Observer,  problem  of  the,  and  three 
objects,  140. 

Oughtred,  30. 


Paciolus,  Lucas,  59,  60. 

Pappus,  59. 

Parabolic  curves,  144  et  seq. 

Peletier,  61. 

Peyrard,  58. 

Physics,  astronomy,  and  mechanics, 
curves  of  errors  in,  136. 

Planetarium,  9. 

Point  in  space,  position  of  a,  139. 

Polygon,  problem  of  the  circle  and 
inscribed,  138. 

Polytechnic  School,  v,  xi. 

Positive  roots,  superior  and  inferior 
limits  of  the,  109. 

Powers,  10  et  seq. 

Practice,  theory  and,  43. 

Present  value,  15. 

Printing,  invention  of,  59. 

Probabilities,  calculus  of,  45  et  seq. 

Problems,  no;  for  solution,  62;  em- 
ployment of  curves  in  the  solution 
of,  127-149. 

Proclus,  59. 

Progressions,  theory  of,  12,  14. 

Proportion,  n  et  seq; 

Ptolemy,  59. 

Radical  expressions  in  equations,  134 

Radicals,  cubic,  75. 

Ratios,  constant,  42  ;  2,  n  et  seq. 

Reality  of  roots,  76,  83,  85,  93. 

Regulafalsi,  137,  148. 

Remainders,  theory  of,  34  et  seq.,  38. 
negative,  35  et  seq. 

Romans,  mathematics  of  the,  54. 

Roots,  negative,  66;  of  equations  of 
the  third  degree,  71  ;  the  reality  of 
the,  74,  76,  79,  83,  85,  93;  of  a  biqua- 
dratic equation,  94;  multiple,  105; 
superior  and  inferior  limits  of  the 
positive,  109;  method  for  finding 
the  limits  of,  no;  separation  of 
the,  112;  the  arithmetical  progres- 
sion revealing  the,  112  et  seq.,  120; 
quantity  less  than  the  difference 
between  any  two,  113;  smallest,  116 
et  seq. ;  limits  of  the  positive  and 
negative,  119. 

Rule,  Cardan's,  68;  of  false,  137;  of 
mixtures,  44  et  seq.;  of  three,  II  et 
seq.,  40  et  seq. 


i56 


INDEX. 


Science,  history  of,  22;  development 

of,  vii  et  seq. 

Seven,  tests  of  divisibility  by,  35. 
Short-mind  symbols,  vii  et  seq. 
Signs  +  and  — ,  57. 
Squaring  of  the  circle,  62. 
Stenophrenic  symbols,  vii  et  seq. 
Straight  line,  144. 
Substitutions,  in  et  seq.,  123. 
Subtraction,   new   method   of,  25   et 

seq. 
Sum  and  difference,  of  two  numbers, 

56. 

Supposition,  rule  of,  137,  148. 
Symbols,  vii  et  seq. 

Tables,  137;  expedient  for  supplying 

lacunae  in,  148. 

Tannery,  M.  Paul,  58  footnote,  151. 
Tartaglia,  60,  61. 
Temperament,  theory  of,  23. 
Theon,  59. 
Theory  and  practice,  43. 


Theory  of  remainders,  utility  of  the, 

38- 

Third  degree,  equations  of  the,  71,  82. 
Three  roots,  reality  of  the,  93, 
Trial  and  error,  rule  of,  137,  148. 
Trisection  of  an  angle,  62,  81. 
Turks,  58. 

Undetermined  quantities,  82. 
Unity,  three  cubic  roots  of,  72. 
Unknown  quantity,  55. 

Values,  mean,  45   et   seq.;    minimal, 

132. 

Variations,  calculus  of,  x. 
Vatican  library,  58. 
Vieta,  viii,  62,  65. 
Vlacq,  20. 

Wallis,  viii. 

Wertheim,  G..  58  footnote. 

Woodhouse,  x. 

Xylander,  58. 


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